The paper presents a technical station model using the graphical evaluation and review technique – GERT. The network model reflects the operation of trainset service in compliance with the technology and technical equipment as well as interaction between technological systems at the service stations that are different by track development but connected to big passenger stations, which gives a possibility of determining the service time of trainsets depending on the number of carriages and the specificity of operation conditions. This GERT model is applied to the case study of an individual service station within the railway network in Bulgaria. The regression functions of service time are worked out according to the type and the number of carriages taking into account the limitation parameters. The functions obtained could be used for trainset scheduling to optimize the operation of passenger carriage fleet.

The complex of passenger and technical stations is one of the railway transport subsystems, which are most complex and difficult to describe and present. The passing and service of coach flows of different types (trains, trainsets, or coach sets subjected to repair or equipment) load railway subsystems in a different way. Due to the complex organization of coach flows maintenance, the time spent at technical stations is a considerable part of the total duration of stay. Waiting for service due to congestion of arrivals and/or lack of capacity results in increase in the duration of interoperation stays. The different options to organize coach flows at a technical station lead to:

change in the duration of stay for equipment operation at a technical station;

change in trainset scheduling, which is basic for scheduling plan and train traffic schedule in the railway network.

Hence, technical stations are complex technological sites where the processes of carriage handling and maintenance are discrete and stochastic, and the way of defining the duration of trainset stay should consider these peculiarities.

References [

The analysis of references has shown that different approaches and criteria are used to determine the duration of trainset stay at technical stations for technological operations. One of the suggestions to determine the duration of this stay is based on regulations where its determination depends on the train category [

Another method used to determine the duration of trainset stay at a technical station is based on the regulations of carriage service and shunting and depends on the number of tracks and operation technology. The duration of technical service at a given yard is determined on the basis of the number of carriages in the set. Thus, the duration of service is determined on the basis of the above-mentioned regulations that vary from 3 to 5 hours per set, depending on the number of carriages included in each set [

The determination of the duration of trainset stay for technological operations both in the passenger and technical stations can also be done using the network planning method [

The total number of coaches in a trainset is determinative factor for both the nature of technological maintenance and the duration of stay necessary, as well as the labor costs of service provided during the determined stay of trainsets.

All three methods mentioned above are used to determine the time duration. The duration of service and interoperation stays (times of waiting) are of stochastic nature, and to determine them, we shall determine the duration of trainset stay at a technical station.

Another problem with coach set service in the complex of passenger and technical stations is shunting for translocation of sets [

It is important to analyze the yard cost parameter, too. It is the time cost saved by each car while passing through a technical station without reclassification [

To present the station and determine the length of the technological processing operations, it is possible to use “queuing theory” [

Studying network algorithms and programs, it has turned out that the network models with a stochastic structure are more flexible and useful than the determined networks. Therefore, the method that should be used has to take into account these requirements, so we have chosen the graphical evaluation and review technique (GERT) networks and GERT systems [

The stochastic GERT network is defined as a network that can be realized only on the basis of certain subsets and arcs. The realization of each branch/operation is selected according to probability distribution. For the realization of a node in stochastic models, it is not necessary to realize all the branches included in the node. Therefore, cycles and loops are allowed in these models.

Let us examine G = (N, A) network containing only GERT nodes, which form a set of N. Let the performance duration of operations (i, j) be random value Yij. A certain operation (i, j) can be performed only in case when node i is realized. In order to study the questions related to the realization of this operation, it is necessary to know the conditional probability (in a discrete case) or the distribution density (in a non-discrete case) of random quantity Yij under the condition that node i is realized. This allows us to carry out a study on the implementation of all networks. The aim of using a GERT network in the stochastic network analysis is to determine the mathematical expectation and dispersion of time to pass through the network (considered here as a general network parameter). Obviously, the coefficient of GERT branch traverse has a corresponding W-function.

The Laplace–Stilt transformation is used to determine the derivative function and transformations needed. The conditional derivative function of random quantity Yij moments is defined as follows:

For non-discrete random variables:

For discrete randomvariables:

In particular, _{ij} (s)^{sa}^{sa}_{ij}

If a=0, to _{ij} (s)

Let Pij be the probability of the performance duration of operation (i, j) under the condition of i node realization. W-function for random variable Yij is determined as follows:

Using transformations, it is always possible to define the G’ network structure, which is identical to the G network structure, where there is only one parameter Wij instead of the two parameters of branches Pij and Yij. If the realization duration of operations in G network is represented as an independent random variable, then G’ has a number of properties that are of certain interest from a computational point of view. To explore these properties, three particular cases are considered:

G’ consists of two or more consecutives: W-function of equivalent branch is equal to the product of W-functions of the consecutive branches.

G’ consists of two or more parallel branches: W-function of equivalent branch is equal to the sum of W-functions of the parallel branches.

G’ consists of a branch and a loop and is reduced to a single branch equivalent to it, for which W-function is equal to the following formula:

If the GERT network consists of parallel and series chains and/or loops, it can be converted into an equivalent network consisting of only one branch. In fact, the result generalizes any GERT network, because the known basic transformations can be combined.

The main steps of using a GERT network are:

to present the network as a stochastic network with GERT nodes;

to calculate W-function for any other network; and

to transform a network into an equivalent network consisting of only one branch.

To transform a network of any structure into an equivalent graph with a much simpler structure, most often Mason’s topological equation is used having the following appearance:

where

To determine the equivalent coefficient of traverse of all loops of m order.

To sum up the variables obtained on all loops of m order and to multiply the result by (-1)^{m}. For a loop of even order variable (-1)^{m} is positive, and for a loop of odd order it is negative.

To add step 1 to the expression obtained from step 2 and equal the result to 0.

To apply (_{A}(s), which joins chain end _{A}(s) function is needed in order to find the equivalent W-function for the output network. In Fig. _{E}(s).

Equivalent function

The given closed graphflow contains a loop of the first order with an equivalent coefficient equal to W_{A}(s).W_{E}(s). Using Mason’s rule, it is obtained that 1-W_{A}(s) W_{E}(s)=0 or W_{A}(s)=1/W_{E}(s).

It should be noted that the topological equation contains function W_{A}(s), since it is an element of a loop of the first order. The significance of the result obtained with examining the above-mentioned example is due to the fact that if W_{A}(s) in the topological equation is replaced with 1/W_{E}(s) and is solved in regard to W_{E}(s), an equivalent W-function for the output stochastic network will be obtained.

To determine the equivalent W-function of an open stochastic network, the following steps are required:

to close a given network with a branch coming out from the end node and entering the initial node;

to find all loops of order m; and

to apply topological equation (_{E}(s).

The expression for equivalent W-function of W_{E}(s) network was obtained from topological equation (_{E}(s)=1 with s=0.

When W_{E}(s)=P_{E} .M_{E}(s), P_{E}= W_{E}(0), from where it follows that:

It should be noted that W_{E}(s) could be expressed by W-functions of all or some of the output network branches. It is not difficult to calculate the amount of W_{E}(0); therefore, s=0 must be set in the expression for W_{E}(s) derived from formula (

Computing the j-partial derivative by s-function ME(s) and laying s=0, jth first μjE of the random variable is found out, i.e.,

In particular, at s=0, the first moment µ_{1Е} represents the mathematical expectation of network realization time where the dispersion of network realization time is equal to the difference between the second moment µ_{2Е} and the square of mathematical expectation µ_{1Е}, i.e.,

To determine the duration of trainset stay using GERT netsworks, an algorithm is developed following the steps given in Fig.

Algorithm of determining the duration of trainset stay at a station

The operation technology at the station serving trainsets is presented as a stochastic network with GERT nodes in regard to the track development and operations.

The examined length of technological processes is a continuous random variable, which is set either with a distribution function (integral distribution law) or with the density of probabilities (differential distribution law). To determine the distribution laws of technological times, it is necessary to make significant statistical observations on the duration of each technological operation.

Determination of the probability and time featuress of «B» branches.

3.1. Finding the probabilities of GERT network arcs in the classical way.

3.2. Determining the type of law for distribution of service times for technological operations using software for statistical data processing or another selected method.

The function of traverse of GERT network branches has a corresponding W-function, and it is determined depending on the composed stochastic network with GERT nodes presenting the station operation technology and then converted into an equivalent network consisting of only one branch.

The aim of using a GERT network is to determine the mathematical expectation and dispersion timing of the network. After determining the probabilities of realization of each arc and defining the law ofdistribution of service times for individual operations, the differential function of distribution f (x) is used with its parameters as given in Table

Parameters of distribution functions

Distribution law | Distribution density of probabilities f (x) | Function causing moments M(x) | First moment – mathematical expectation E(x) | Second moment – dispersion D(x) |
---|---|---|---|---|

Normal | a | a^{2} |
||

Gamma | a.b | a.b^{2} |

In order to determine the performance indicators of the station, using a GERT network, the following cases are considered: when there are two normal distributions; one normal and one gamma distributions, and two gamma distributions. Using the usual attributes of the exponent and the assumed substitutions, the equivalent function is deducted expressed by the first moment, mathematical expectation Е_{ij}(X), the second moment, dispersion D_{ij}(X), and realization probability of the respective branch P_{ij}. It has been proved in [

The developed algorithm has been applied to Nadezhda technical station, which serves Sofia passenger station. The main task of Nadezhda–Sofia technical station, the largest one in Bulgaria serving 75% of all coaches, is to provide roadworthy, clean and equipped railway vehicles for safe and comfortable travel.

The main technological operations carried out at a technical station are [

Technical inspection: it starts as soon as the shunting trainset passes through the switch crossing, where it meets the two inspectors standing on its both sides. The trainset is checked for sounds in order to detect faults on the bogie or in the automatic braking system. Any observed malfunctions, which can be removed without the detachment of the coach from the trainset, are taken care of in the workshop fully equipped for this purpose.

The coaches with malfunctions are removed from operation, detached from the trainset, and lined on a track for repair. The coach taken out of the trainset is replaced with a repaired coach available. When necessary, a change of the trainset is made.

Carriages are cleaned daily – internal cleaning of all types of carriages.

Main internal and external cleaning of certain trainsets depends on the category of trains to be served in a certain time interval. External washing is done by pushing the trainset through a washing machine at a speed of 2 km/h.

Technical maintenance and servicing of equipment of the trainsets are carried out in the workshop for the repair of coaches on one of the tracks for equipment.

When a set of coaches are ready for departure to the passenger station, a shunting locomotive is delivered to it in the equipping workshop or on the arrival–departure track where the set is stored after being prepared.

The track development of any technical station determines the technology of handling the trainsets. The work of the repair and equipping and the carriage and repair depots and inspection posts depends on the track development schemes and the number of tracks. The model of Nadezhda technical station has been developed as a stochastic network with GERT nodes, considering its track development and sequence of the operations performed and shown in Fig.

The GERT model of a technical station

W_{12} – performing technical inspection, waiting and performance of internal cleaning in RP;

W_{13} – performing technical inspection and waiting and performing internal cleaning on the equipment tracks;

W_{24} – moving the trainset and passing through an external washing device;

W_{25} –technical maintenance and equipment on the equipment tracks;

W_{22} – due to the malfunction of any coach, it is replaced with an available one by shunting;

W_{35} – technical maintenance and equipping on the equipment tracks;

W_{33} – due to malfunction of any coach, it is replaced with an available one by shunting;

W_{45} – technical maintenance and equipping on the equipment track;

W_{56}^{1} – storage on the equipment tracks;

W_{56}^{2} – moving from the equipment tracks to the arrival–departure one;

After determining the distribution laws of service times and numerical parameters, the probability of each arc is determined depending on the group under consideration and the collected statistical information. The results are shown in Table _{ij} arcs described in Section 4, Step 1, as the results for each arc are given for selected groups under examination.

Parameters of arcs

Arcs | Probability P(x) | Distribution law | Mathematical expectation E(x) | Dispersion D(x) | Average Quadratic deviation | Variance coefficient-v(x) | P-Value |
---|---|---|---|---|---|---|---|

W12 | |||||||

2 coach | 0,683 | Gamma | 27,412 | 166,382 | 12,899 | 0,471 | 0,374 |

5 coach | 0,227 | Normal | 50,985 | 221,733 | 14,891 | 0,292 | 0,219 |

8 coach | 0,140 | Normal | 69,190 | 282,450 | 16,806 | 0,243 | 0,053 |

11 coach | 0,383 | Normal | 25,906 | 158,404 | 12,586 | 0,486 | 0,261 |

W22 | |||||||

2 coach | 0,013 | ||||||

5 coach | 0,108 | Normal | 7,417 | 3,679 | 1,918 | 0,259 | 0,127 |

8 coach | 0,178 | Normal | 9,331 | 3,952 | 1,988 | 0,213 | 0,465 |

11 coach | 0,156 | Normal | 10,021 | 3,973 | 1,993 | 0,199 | 0,256 |

W24 | |||||||

2 coach | 0,518 | Normal | 82,547 | 346,400 | 18,612 | 0,225 | 0,147 |

5 coach | 0,652 | Normal | 99,087 | 391,600 | 19,789 | 0,200 | 0,072 |

8 coach | 0,744 | Normal | 126,964 | 452,440 | 21,271 | 0,168 | 0,690 |

11 coach | 0,302 | Normal | 219,000 | 680,330 | 26,083 | 0,119 | 0,827 |

W45 | |||||||

2 coach | 1,000 | Normal | 7,500 | 3,364 | 1,834 | 0,245 | 0,223 |

5 coach | 1,000 | Normal | 11,603 | 8,449 | 2,907 | 0,251 | 0,288 |

8 coach | 1,000 | Normal | 12,903 | 9,024 | 3,004 | 0,233 | 0,187 |

11 coach | 1,000 | Normal | 13,053 | 15,053 | 3,880 | 0,297 | 0,540 |

W25 | |||||||

2 coach | 0,469 | Normal | 90,417 | 422,992 | 20,567 | 0,227 | 0,513 |

5 coach | 0,240 | Normal | 123,724 | 495,831 | 22,267 | 0,180 | 0,073 |

8 coach | 0,078 | Normal | 110,625 | 489,583 | 22,127 | 0,200 | 0,432 |

11 coach | 0,542 | Normal | 127,639 | 517,837 | 22,756 | 0,178 | 0,645 |

W13 | |||||||

2 coach | 0,317 | Normal | 8,632 | 25,023 | 5,002 | 0,580 | 0,090 |

5 coach | 0,773 | Normal | 26,576 | 94,648 | 9,729 | 0,366 | 0,094 |

8 coach | 0,860 | Erlang | 48,379 | 195,280 | 13,974 | 0,289 | 0,113 |

11 coach | 0,617 | Gamma | 17,667 | 87,846 | 9,373 | 0,531 | 0,189 |

W33 | |||||||

2 coach | 0,009 | 5,200 | 2,100 | ||||

5 coach | 0,043 | Normal | 6,842 | 4,268 | 2,066 | 0,302 | 0,143 |

8 coach | 0,097 | Normal | 8,248 | 4,295 | 2,072 | 0,251 | 0,547 |

11 coach | 0,117 | Normal | 9,321 | 6,531 | 2,556 | 0,274 | 0,326 |

W35 | |||||||

2 coach | 0,991 | Normal | 79,750 | 259,930 | 16,122 | 0,202 | 0,212 |

5 coach | 0,957 | Normal | 81,346 | 387,115 | 19,675 | 0,242 | 0,541 |

8 coach | 0,903 | Normal | 105,625 | 489,583 | 22,127 | 0,209 | 0,172 |

11 coach | 0,883 | Normal | 138,929 | 506,217 | 22,499 | 0,162 | 0,099 |

W1,56 | |||||||

2 coach | 0,000 | 0,000 | 0,000 | ||||

5 coach | 0,894 | Normal | 41,771 | 189,711 | 13,774 | 0,330 | 0,736 |

8 coach | 0,147 | Gamma | 24,773 | 110,470 | 10,510 | 0,424 | 0,242 |

11 coach | 0,058 | Normal | 36,714 | 135,569 | 11,643 | 0,317 | Mалко данни |

W^{2},56 |
|||||||

2 coach | 1,000 | Erlang | 26,189 | 124,694 | 11,167 | 0,426 | 0,494 |

5 coach | 0,106 | Normal | 20,354 | 122,284 | 11,058 | 0,543 | 0,055 |

8 coach | 0,853 | Normal | 19,163 | 126,044 | 11,227 | 0,586 | 0,081 |

11 coach | 0,942 | Gamma | 32,706 | 124,972 | 11,179 | 0,342 | 0,259 |

Equivalent function W_{16} depending on the scheme in Fig.

Formula 3.2.11 is used to close the chain:

K1256 and K1356 are applied

The equivalent equation appears as follows:

In order to find the dispersion of operation time, it is necessary to find the second derivative of M16(s) in a similar way, and the dispersion itself at s=0 is equal to the difference between the values of the second derivative and the square of first derivative (formula

To simplify the procedure for determining the mathematical expectation and dispersion of the duration of trainset stay at the technical station, an Excel computational file has been developed where the variables sought are determined by entering probability of the given arc and the two distribution moments.

Using the developed computational procedure, the features of selected groups for the GERT network are determined – the mathematical expectation and dispersion of the duration of trainset stay at the technical station given in Table

Features of selected groups of carriages

2 coach | 5 coach | 8 coach | 11 coach | |
---|---|---|---|---|

Duration of the trainset average stay (minutes) | 135 | 144 | 183 | 202 |

Dispersion of the trainset average stay | 737 | 1150 | 1450 | 1611 |

Average square deviation (minutes) | 27 | 34 | 38 | 40 |

Using the integral function of normal distribution, the cumulative function given in Fig.

Cumulative function of times for service of the selected groups

The average value plus the average square deviation involving 84% of the cases examined has been assumed as a standard of the duration of trainset stay. However, this would not reduce the reliability of the normal handling of trainsets because the rest of cases with longer stays are due to the inability to separate all trainset stays due to the set-up schedule. The standard of the duration of trainset stay Tdt (n) depends on the number of carriages in the trainsets to be served – . An exponential function has been chosen to approximate the assumed standards, as approximation is practically implemented through add trendline of Microsoft Excel and Microsoft Office.

where n is the number of carriages in the set of coaches/ trainset; R^{2} = 0.98 – coefficient of determination.

The rules used for determining the duration of trainset stay according to the number of coaches can be used to schedule trains in compliance with the passenger train traffic schedule.

Having defined the functions of different types of stations, their standardization could be included as regulations in references [

The analysis of the methods used to determine the duration of trainset stay at a technical station has led definition of problems, objectives and choice of an appropriate methodological approach to solve them. The algorithm developed for determining the duration of trainset stay using GERT networks is universal regardless of the type of station, which it is applied to.

The developed network model with a stochastic structure allows us to reflect the complex interaction of technological systems and to present stations arbitrary in schematic development. The model makes it possible to determine the duration of trainset stay according to their size and the technical equipment and operating technology at the stations under examination. The developed model (Fig.

In order to realize the model, it is necessary to provide a database of statistical information about the technological processes performed and determine the laws for the distribution of the service times and probabilities of the arc conditions. Appropriate regression functions of the duration of the service of coach/trainsets have been found depending on the type and number of carriages at technical stations taking into account technological features and technical constraints affecting their type (Figure

The general model of station as a network model is valuable at the level of object theoretical description. It makes possible to get specific results that can be used successfully for practical applications. The Nadezhda–Sofia station is the largest technical station in the country and the created model of stochastic network with GERT nodes as given in Figure

This model can be used also for other technical areas by dropping some of its nodes corresponding to the state of the stochastic network taking into account the technical equipment. For example, the technical region of Plovdiv node 3 in the model should be dropped because there is no direct handling to the equipment rails, and the region of Varna node 2 should also be dropped because washing of the trains is carried out either in Nadezhda technical station or in the technical region of Plovdiv. The railway network in Bulgaria has many deviations and the nodal sectional stations, where the connection with main lines is made as well as the terminals of main lines are the end stations for passenger trains used to perform some of train servicing operations. The change of the stochastic network model with GERT-nodes for the two technical regions mentioned above or terminals leads to the simplification of equations for the first and second moments (formulas 17, 18) and, hence, finding out mathematical expectation and the dispersion of the network realization time (formulas 8, 9) is simplified too. Therefore, it can be said that the use of this model is universal in terms of determining the downtime of passenger trains at each station, which is the end destination of arriving trains and the place where these trains must be serviced. The model can also be applied to determine the minimum time of scheduling at each station regardless of the country where it is used due to the almost identical technological processes of passenger train servicing, as the stochastic network with GERT-nodes (Fig.

The report reflects the results of project “Methods for Optimization of Railway Suburban Transport” funded by Todor Kableshkov University of Transport.