Mathematical modelling of the railway station ’s technological parameters in transport corridor system of export traffic increasing volumes

. The study substantiates the need to develop mathematical analysis methods of railway station technology in the context of increasing the volume of freight traffic through the constituent elements of the infrastructure of the international transport corridor. The aim of the work is to develop proposals for improving the station operation technology under the conditions of its additional functions, to analyse the unevenness of the station transportation process and to assess changes in the transport corridor parameters. The object of the study is the Astrakhan-II station of the Volga Railway (JSC "Russian Railways"), located within the boundaries of the international corridor "North-South". In the article on the example of a particular station, its technical and operational characteristics, time norms for the performance of various technological operations, the unevenness of the station process is assessed, and a mathematical model of the station operation as a mass service system is developed. The "bottlenecks" are identified and proposals are formed to increase the processing capacity of the station, aimed at reducing channel utilisation and downtime duration. The results of the mathematical model are supplemented with technological proposals to change the formation plan of the Astrakhan II station to assist the port of Olya in picking up ship consignments in order to maximise the efficient use of Astrakhan II station capacity and reduce the need for track development in the port of Olya for marshalling operations, and consequently reduce the cost of its development. As part of compliance with the coordinated development of all elements of transport infrastructure, the capacity of the railway approaches to the station is assessed and a final set of measures is established.


Introduction
The relevance of the topic is the change in the volume of freight traffic to and from South-East Asia due to the current reorientation of the main export cargo flow and changes in political and economic relations of a number of countries, which requires planning and design of railway corridors within the boundaries of these changes.In recent studies, the problems of developing new and reconstructing existing transport corridors are considered from the point of view of improving their sustainability, ensuring a given volume of traffic or its maximisation, and taking into account the criteria of environmental and geological sustainability [1][2][3][4][5][6].Thus, in [2], using the example of the Baikal-Amur Mainline, the main attention is focused on ensuring the rhythm and stability of traffic by increasing the capacity of single-track railway sections.In [3], the emphasis is placed on the tasks of modernisation of mixed-use railway networks, primarily in the development of passenger traffic.This study sets the task of analysing the stability of railway station operation to accommodate the growing volumes of freight traffic in the transport corridor.The efficiency of the development is determined by the quality of the organisation of freight traffic passing through the constituent elements of the transport infrastructure of the direction under consideration.Such elements are the Astrakhan-II station, which gets the status of a marshalling station to accelerate the processing of transit wagon traffic along the North-South corridor, and the approaches to it.The task is complicated by the fact that the station is assigned new functions to handle additional wagon traffic for the port of Olya.Therefore, within the framework of the research the tasks are set to assess the capacity of the Astrakhan -II station to process additional car traffic, to identify bottlenecks that will not allow to master the forecasted increase in the volume of work, to form scientific and technological proposals for their elimination.Thus, the structure of this scientific research includes setting the research objectives, analysing the mathematical basis of the research, selecting the most appropriate mathematical apparatus for solving the tasks set in the research, defining the final set of measures.

Input data for the calculation model
The following are considered as input data: 1) forecast volumes of freight traffic on the approaches to the station and port (Verkhny Baskunchak -Aksarayskaya -Astrakhan-II -Olya (Fig. 1), million tonnes, which determine the intensity of incoming train traffic to the station.2) the size of processed wagon flows by destination.According to the train formation plan, Astrakhan-II station forms 12 appointments, including: 8 appointments of through trains, 1 assembly train (1 train per day, 71 wagons), appointment of transfer trains (2 trains per day, 57 wagons), 2 appointments of outbound trains.Total processing carload traffic -1404 cars.3) Technological norms of operations and technical and technological equipment of the station, including locomotives, slide, teams of inspectors and wagon drivers, which determine the intensity of train flow service.

Estimation of unevenness of train arrival and service at the station and establishment of the theoretical law of distribution of random variables
Significant impact on the performance of the plant is: -uneven arrival of trains for dismantling, which manifests itself in the variation of intervals between the moments of arrival of dismantling trains (T inlet ) and, as a consequence, in the change in the intensity of the incoming flow (formula (1)) -non-uniformity of formation, technical inspection, provision of locomotives, and dispatch to sections, which manifests itself in the variation of the duration of train processing in these systems (T serv ) and as a consequence, in the change of service intensity (formula (2)) where K defines the number of channels.
Considering a railway station as a queue system (QS) [7,8], it is possible to formulate the basic condition of interaction: the service intensity should be greater than the intensity of the incoming flow entering the service system (formula (3)) In order to assess the fulfilment of this condition, statistics of such values as intervals between train arrivals, train inspection duration, and disbanding duration were collected and processed, the results of which are presented in Tables 2 -4.
The empirical plot of the distribution density of these values is presented as a histogram (Figures 2, 3, 4), the ordinate of which is obtained by dividing the frequencies by the values of the corresponding discharges (formula (4))): Based on the constructed histograms, a visual assessment is made of the conformity of the statistical distribution to one of the theoretical distribution laws, which are used for further calculations.
In addition, the deviation from the mean is expressed in a more convenient form as: -squared deviation (formula (5)): -coefficient of variation (formula (6)): Based on these characteristics, a conclusion is made about the degree of unevenness of the values under consideration.For the remaining operations included in the model, for which there is no statistical data processing, the duration of operations with regard to irregularity is corrected according to the formulas using the network average values of the coefficients of irregularity of operations execution.
Having analysed the data obtained, the following conclusions can be drawn: -the degree of unevenness of train arrival intervals at the station is high, as the coefficient of variation is 0.9 and is close to 1; -the degree of unevenness of technical inspection and disbanding is low, as the coefficient values are close to zero and have values of 0.3 and 0.28, respectively; -the exponential law of probability distribution can be used to describe the incoming flow; -intensity of the incoming flow according to formula (1) is equal to  = 37 train/day; -intensity of train maintenance during technical inspection in accordance with formula (2) when two crews are working is equal to µ = 52.8train/day, during disbanding the intensity of train service is equal to µ = 39.44 train/day; -when trains arrive at the station for processing, condition (3) is fulfilled under the existing workloads; to assess station operation in the future, a station operation model needs to be developed.

Development of a model of railway station operation
The performed analysis of operational characteristics indicates changes in the station operation volumes.In the future, it is expected that the volume of work will increase.In this regard, it is necessary to develop a model that allows to evaluate the existing technology of station operation and, when changes are made in the volume of work performed, to give recommendations for adjusting the number of crews, locomotives and the order of servicing objects.
In modelling the station operation [7,8] it is proposed to use a mass service system, where trains and local transmissions are considered as requests, and its fleets and technical devices are considered as a servicing device.
The purpose of modelling is to establish the real idle time of wagons at the station, taking into account the unevenness of technological operations, leading to fluctuations in the duration of their performance and the occurrence of idle time waiting for these operations [9].Based on this, the load of station process elements is determined and the required track The model of wagon progression through the station is presented in Figure 5.
As previously stated, the results of the modelling depend on the number of operations to be performed, determined by the number of incoming trains and the number of local transmissions moved from one fleet to another, as well as the distribution of their processing between the different channels.Based on the operational characteristics and technology of the station, Table 5 gives an example of existing workloads for the service channel "carload carriers of Park "A" and "B".Similarly, the following service channels are investigated: "car drivers of the "B" fleet", "shunting locomotives", "train drivers", etc., the values for which are entered into the developed model of wagon progression through the station (Figure 5).non-uniformity according to formula (7) for shunting operations (except for the interval of train disbanding on the slide) and formula (8) for other station technological operations.

𝑀(𝑡
where  ℎ -technological standard;    -the coefficient of unevenness of operation performance.

𝑀(𝑡
On the basis of point 3, taking into account non-uniformity of station processes, the elements of service by channels and duration of service of this element are corrected, for example, Table 6 shows the corrected values of duration of shunting operations.Based on this information, as well as information on the number of operations performed, the following characteristics are calculated for each system, which are necessary to determine train idling and the number of trains at the station:  -Intensity of trains entering each system (corresponds to the number of trains/transfers processed per day);  -intensity of trains leaving the system under consideration (formula (2))  -system load, equal to the ratio (formula ( 9)): In calculating  it is necessary to take into account that the channel servicing the system under consideration may be diverted during the day to service other systems according to Table 5.In this case, the allocated time during the day to service the system under consideration is reduced by the amount required to service other systems.Then formula (10) will have the form When the system is served by several channels, the formula will be as follows If channels serve several systems, in order to check the condition that their load does not exceed permissible values, it is necessary to determine the total load of the channel by adding all time costs of operations performed by the channel under consideration (formula (11)): The results of the channel load test for the existing train traffic are shown in the table below 7.If all technological lines are loaded, taking into account the unevenness of the technological process within the permissible norms, it is concluded that the station is able to absorb the entire train flow and no measures for technology improvement are required.For the considered example of Astrakhan-II station, no improvement measures are required at the existing volumes of work.However, the load of individual technological lines has a maximum permissible value (0.886) and the station will not be able to cope with an increase in the volume of work.In this regard, measures are required to increase the intensity of train service, which include automation of operations, for example, by introducing such systems as ACS TIS (automated control system of the technical inspection station), SALS (shunting automatic locomotive signalling) and an increase in the number of technological lines.

Estimation of element utilisation after implementation of the proposed measures and increases in workload and establishment of plant operating model indicators
The intensity of train arrivals is based on the projected volumes that were presented in paragraph 2.
The duration of operations has been reduced taking into account their automation.
The results of loading of technological lines for evaluation of the proposed measures are presented in Table 8.The load values obtained indicate that, with the implementation of the proposed measures, the station will be able to process the predicted train flow; therefore, it is possible to assess its performance by calculating the station operation model indicators.Since it was established in point 3 that the unevenness of train arrivals at the station is described by an exponential law, the Erlang formulae can be used to calculate the performance indicators of the station's mass maintenance systems.: -average number of trains in the queue -average dwell time of trains in the system The results of calculation of indicators of the proposed station operation model for the technological line "Wagoners in park "A" and "B" are presented in Table 9.The calculation of indicators for other technological lines and operations is performed in the same way.
Based on the number of trains in each system of Astrakhan-II station, the adequacy of track development is established (see Table 10) According to this table, if the proposed measures are implemented, the track development does not need to be expanded even if the existing track development is increased.In order to effectively utilise the capacity created at the station, it is necessary to ensure that the expected train traffic also passes through the approaches to the station.In order to comply with the principle of coordinated development of all elements of transport infrastructure, it is necessary that the capacity of the approaches corresponds to the capacity of the station.Therefore, for further studies on the sustainability of transport corridor elements, it is necessary to assess the capacity of the directions, using the already available methods of assessing the capacity of railway lines, as reflected in the studies [4,[10][11][12][13][14][15][16].

Conclusion
The study uses the example of a specific railway station in the North-South corridor, taking into account its technical and operational characteristics, time standards for performing various technological operations:  assessed existing station technology in the context of projected traffic growth;  The nonuniformity of the station process is assessed, a mathematical model of station operation as a mass service system is developed;  identified bottlenecks that will delay the passage of train traffic and reduce the sustainability of the transport corridor.; provides scientific and technological solutions to remove bottlenecks,  the proposed technology was evaluated under conditions of increasing traffic volumes and the following tasks were formulated to assess the sustainability of transport corridor operations.

Fig. 1 .
Fig. 1.Forecast volumes of cargo flows at the approaches to the station and the port of Olya, mln tonnes.

Fig. 4 .
Fig. 4. Histogram of the distribution of intervals of train disbanding durations.Obtained probabilistic characteristics:   = 10.12;  = 0.28; () = 36.51min; () = 102.38.For the remaining operations included in the model, for which there is no statistical data processing, the duration of operations with regard to irregularity is corrected according to the formulas using the network average values of the coefficients of irregularity of operations execution.Having analysed the data obtained, the following conclusions can be drawn:-the degree of unevenness of train arrival intervals at the station is high, as the coefficient of variation is 0.9 and is close to 1;-the degree of unevenness of technical inspection and disbanding is low, as the coefficient values are close to zero and have values of 0.3 and 0.28, respectively;-the exponential law of probability distribution can be used to describe the incoming flow;-intensity of the incoming flow according to formula (1) is equal to  = 37 train/day; -intensity of train maintenance during technical inspection in accordance with formula (2) when two crews are working is equal to µ = 52.8train/day, during disbanding the intensity of train service is equal to µ = 39.44 train/day; -when trains arrive at the station for processing, condition (3) is fulfilled under the existing workloads; to assess station operation in the future, a station operation model needs to be developed.

Fig. 5 . 5 . 7 6 7 FO in Park B 7
Fig. 5. Model of car traffic advancement at Astrakhan-II station.Symbols: PO -preparatory operations, TI -technical inspection, CI -commercial inspection, TD -transport documentation, WG Acc.-wagon guarding, LH -locomotive hitching, BTbrake test, DI -issuance of transport documents to the locomotive crew, FR -fixing removal, FO -final operations, DISB -disbanding to the railway station -to the access road, Car.Oper.-cargo operations, C -coupling, Ddisconnection, BLC -brake line charging, FBT -full brake test, FR -fence removal, ABT -abbreviated brake test, DISP.-dispatchTable 5. Distribution of workloads among the elements of the plant operations model.Service channel No. Element QS Packs train/front

-;-
the probability that there are n trains in the system the average number of trains in the system, which includes both queues and facilities E3S Web of Conferences 431, 08014 (2023) ITSE-2023 https://doi.org/10.1051/e3sconf/202343108014L k = L S + λ ef μ average length of time trains stay in the queue λ

Table 1 .
Size of wagon flows by destination.

Table 2 .
Characterisation of train arrival intervals.

Table 3 .
Characterisation of the duration of the technical inspection.

Table 4 .
Characterisation of the duration of train disbanding.

Table 6 .
Duration of operations performed at the station.

Table 7 .
Calculation of system utilisation by aggregated elements.

Table 8 .
Estimation of station element utilisation when the workload increases to the projected workload.

Table 9 .
Indicators of mass maintenance systems of Astrakhan-II station after implementation of measures and increase of work volumes.

Table 10 .
Analysing the conformity of actual track development with the needs.