LENGTH OF TRANSFER TIME BETWEEN CONNECTING TRAINS – IS THERE AN IDEAL VALUE?

. Quality public transport is an essential part of global mobility. In general, there is an effort to ensure that as many transport users as possible use public transport. The public transport subsystems also include rail transport. Nowadays, traveling by train is not only a matter of overcoming longer distances in national and international transport, but also a matter of daily commuting to work, school or service. Suburban rail transport is usually part of integrated transport systems, the principle of which is the possibility to use several types of means of transport in one journey per ticket. An indicator of the proper functioning of these systems is the existence of quality and convenient transfer dates times and reliable transfer links that minimize waiting times and total time spent on transportation. Within these systems, there is no network-specific maximum transfer time, which usually depends on the technical solution of the transfer point, the distance that passengers have to cover between vehicles or check-in technology (for example, front door boarding, turnstiles, etc.). the ideal value of the transfer time in terms of the probability that the passenger will not miss the connection and at the same time represents different aspects of the view of this issue. This is on the specific route Ostrava – Prague, which is the busiest in the Czech Republic. Within it, two reference links will be selected (one for each direction), the mean value of the delay in the stations with connections and other statistical indicators will be determined. Furthermore, a simulation of train delays is created using the Monte Carlo method, from which the probability of a connection is determined and subsequently the degree of dissatisfaction with the route is determined while maintaining the current connections. From which a transfer time is subsequently identified in which the probability of passing is as low as possible, but at the same time in normal situations passengers do not have to wait too long.


Literature Review
According to Higgins et al. [1], waiting time is a critical aspect for passengers in deciding on the use of public transport and their perception of the quality of the service.During train journeys, there may be delays for certain reasons, which have a significant effect on the quality of public transport, with missing a connection at the planned transfer point being the worst possible option.The study by Vanstenwegen et al. [2] solves the railway timetable in Belgium and, with the help of linear programming, measures its performance by determining the value of the costs associated with waiting for a connecting link at a transfer station.On the selected railway network, the researchers used this method to find a significant reduction in the costs associated with waiting, with a 30 % reduction in transfer time and the resulting overall delay.However, reducing these costs will increase the number of times a trailer misses (by up to 12 %).According to Goverde [3], the term waiting time in rail transport has several meanings, namely the time after the passenger boarding the train before the train starts, the time the train stays at the railway station where the passenger does not leave, the transfer time at the passenger transfer station the train does not miss/miss and then again the time before the start of the train to which the passenger transferred.The total transport time then consists of these individual components of waiting times and have an impact on the quality and competitiveness of public transport.

Solution in the Czech Republic
Within this issue, the authors of the article determine three model situations that are very easily applicable to the environment of suburban and regional railway transport in the Czech Republic.In principle, there are 3 basic intervals between connections for this form of rail transport.A 30-minute interval in the case of suburban transport to a large agglomeration, a 60minute interval in the case of railway lines connecting to the main suburban lines and an interval of 120 minutes on normal regional lines.This value is crucial for the calculation, as it is the basis for how many minutes the passenger will have to wait if he misses the connection due to the delay.Due to the independence of local conditions, no specific route was selected, but a transport time of 60 minutes was set.Subsequently, the values for the transfer are determined, namely 5, 10 and 20 minutes, which increase the transport time accordingly.According to the available statistical yearbook of the infrastructure manager in the Czech Republic [4], specifically the Správa železnic, s.o. it is stated that 90.81 % of passenger trains in 2020 were run on time or with a maximum delay of 5 minutes.
Due to the ease of interpretation, the train connected to the connecting train will be marked first, the connecting train itself will be marked second.The authors assume that the second train at the transfer point will always wait for the arrival of the first train, which is not more than 5 minutes late.This means that the transfer time of 5 minutes will be increased by the time necessary for the transfer of passengers with regard to the amount of delay of the first train.It follows from the above that 9.19 % of trains were operated with a delay of more than 5 minutes in that year.However, this does not mean that in all cases of such a delay the second train will run on time.
The authors assume that in the case of a transfer time of 5 minutes the second train will always run if the first train is delayed by more than 5 minutes, in the case of a transfer time of 10 minutes the second train will run if the first train is delayed more than 10 minutes.The connection is not maintained if the first train is delayed for more than 20 minutes.Data on individual minute percentages of delays are not available, so it is necessary to determine for the purposes of the calculation what percentage of delayed trains was over 10 minutes and how many over 20 minutes.The percentages of likelihood of delay of trains and likelihood of loss connection you can find in Tables 1  and 2.

Case Study
In this case study, two connections (IC 548 and IC 549) were observed on the busiest route in the Czech Republic (Prague -Ostrava) in the period from 23 March  To illustrate the whole situation, the authors de-  cided to graphically illustrate the delays on the IC 548 connection in Figure 2, where the dark green color shows the arrival of a train to the station with a delay of up to 5 minutes, light green up to 10 minutes, yellow up to 15 minutes, orange up to 20 minutes, red up to 30 minutes, dark red up to 60 minutes and black above 60 minutes, and the trains not running are then shown in white.
From the above, it can be seen that train IC 548 performs significantly better than train IC 549, even though it runs part of its route in the morning peak.IC 549, even though it departs at the end of the afternoon peak, managed to depart from its departure station only 43 % of the time with a delay of up to 5 minutes in the period under review.It reached the station Návsí with a delay of 5 minutes only in 3 % of cases.
On the basis of the obtained frequency of delays on arrivals to the stations, it is possible to determine the individual statistical parameters that will be used to build a Monte Carlo simulation model.In particular, the mean value of the occurrence of delays λ.It is also known that train delays follow an exponential probability distribution.For simplicity, the authors decided that for train IC 548 the simulation will be performed only at the stations Ostrava-Svinov, Hranice na Moravě, Olomouc Main Station, Česká Třebová, Pardubice Main Station, Kolín and Prague Main Station, which are important interchanges.For train IC 549, simulations will take place at the same stations, except for the station Praha hl.n., which is the departure station for the train and will be replaced by the station Bohumín.Within each station, 1 000 000 train delays will be simulated.The simulation was carried out on the basis of a script written in Python programming language.
The tables at Figure 3  It can be seen from the graphs at Figure 4 and Figure 5 that the average delay is indeed increasing during the train journey, which points to the fact that the timetable of IC 548 and IC 549 is very unstable and is not able to absorb the resulting delays, which clearly has a detrimental effect on connecting links, and therefore it is evident that these will subsequently lead to an increase in dissatisfied passengers, which may escalate into a passenger exodus from public transport and this problem needs to be addressed more.However, it is necessary to add in one breath that during the period under review there were three closures, namely the Přerov junction, the Ústí nad Orlicí -Choceň section, Kolín -Poříčany, which had a clearly negative impact on train travel.
The tables in Figure 6 show the cumulative probability of train delays on arrival at the monitored stations.This information is very relevant for determining the appropriate changeover time for connecting trains.Within the transfer times, the authors decided to establish three categories -stable transfer link (90-100 % probability of securing a transfer), partially stable transfer link (75-90 %) and unstable transfer link (less than 75 %).The table below then shows the individual transfer links at the stations listed.A connecting train is defined as a connection running a maximum of 30 minutes after the arrival of the first train (at Praha hl.n.station this time was set at 45 minutes) and is not a return connection or a   connection in the same category going to the same destination station.The table shows the station, arrival time, number of the connecting train, time until departure of the connecting train, destination station of the connecting train, direction, stability of the connecting link based on the cumulative probability of train delay, number of the next train going in the given direction (without change), time from departure of the first connecting train to departure of the second train, the risk of missing the connection (this is a twodimensional quantity consisting of the complement of the stability of the connection and the time in minutes until the departure of the next train going in that direction) and the proposed transfer time based on the cumulative probability of train delay, so as to ensure a lower bound of the stability of the connection, i.e. 90 %.
The tables in Figures 7 and 8 show that for train IC 548 all interchanges beyond Olomouc hl.n.station are unstable.The transfer coupling at Kolín station to train R 1302 in the direction of Poděbrady, Mělník and Ústí nad Labem hl.n. is the least stable.The highest transfer risk is then the transfer link at Prague Main Station to train EC 333 in the direction of České Budějovice and Linz, where the stability of the transfer link is only 38.3 % and the next connection takes 4 hours.For train IC 549 the situation is even more desperate.None of the interchanges are stable, even for the last connections of the day.In practice, these connections are obliged to wait for the arrival of the delayed train even with a longer delay, but the whole situation leads to disillusionment for passengers.

Conclusion
The analysis of the interchange links and the subsequent simulation shows that the current form of the timetable of trains IC 548 and IC 549 and the timetable of departures of connecting trains is very unstable, where only 5 out of 34 interchange links can be described as stable and 7 as partially stable, the remaining 22 links are unstable.For some links the probability of catching the connection is less than 50 % (9 cases) or even 25 % (3 cases).In many cases, the proposed changeover times are several tens of minutes higher than the current ones, which is definitely not a sustainable solution.Another issue may be the reintroduction of a fast train segment on the Prague -Ostrava route, which could solve the Ex1-R8 and Ex1-R19 interchanges and reduce the risk of double transfers (Ex1-Kopřivnice-Nový Jičín-Bílovec-Náchodetc.).

Figure 1 .
Figure 1.Numbers of delayed trains according to individual time intervals.

Figure 2 .
Figure 2. Distribution of train delays within individual stops.

Figure 3 .
Figure 3. Number of occurences of delayed trains.
show the results of the fre-quency of delays for trains IC 548 and IC 549 in the simulation.At first glance, it can be seen that there is an increase in the average delay for trains in both directions over the course of the train journey.For a better demonstration of this phenomenon, the Figure4show the frequency of delays for the IC 548 service at Ostrava-Svinov and Praha hl.n stations and for the IC 549 service at Kolín and Bohumín stations.

Figure 4 .
Figure 4. Numbers of delayed trains of train number 548 according to individual time intervals.

Figure 5 .
Figure 5. Numbers of delayed trains of train number 549 according to individual time intervals.

Figure 7 .
Figure 7. Stability of transfers during the way of train number 549.

Figure 8 .
Figure 8. Stability of transfers during the way of train number 548.

Table 1 .
Likelihood of Delay (according to minute intervals).

Table 2 .
Likelihood of loosing connection during different transfer times.
individual delays are categorised.It should be noted that 21 connections of trains IC 548 and IC 549 are listed as cancelled in the Návsí -Bohumín section, but these are connections that do not run according to the valid timetable.