Research on the effectiveness of transportation infrastructure maintenance based on the risk process

. In order to improve the efficiency of transportation infrastructure maintenance, it is proposed to introduce an insurance fund that performs two functions: accumulation of payments with different frequency and cost for performing various types of maintenance; pays for these types of maintenance as necessary. To mathematically describe the state of the insurance fund, it is proposed to use a special type of risk process. This approach allowed us to introduce maintenance performance indicators in the form of resource-cost and financial risks, taking into account the possibility of non-performance of maintenances due to lack of financial resources. A modeling program based on the event approach was created in the MATLAB environment to study these indicators. Computational experiments with the modeling program allowed us to draw conclusions that, in order to minimize the risk values, preference should be given to the option when the shares and frequency of payments to the insurance fund are determined based on the type of maintenance and initial data.


Introduction
Maintenance of infrastructure in railway transport includes the operation and maintenance of the track, the contact network, and the entire infrastructure complex, which includes various automated systems [1,2].In accordance with the Federal law "On security of critical Russian Federation information infrastructure" adopted in 2017, the objects of critical information infrastructure are information systems, information and telecommunications networks, automated control systems and other similar objects.Transport infrastructure also falls under the definition of such objects.
Three types of maintenance for transport infrastructure can be identified: 1) current, 2) emergency, 3) investment.To make these types of maintenance more effective, taking into account the developed system of diagnostics and monitoring [3,4], they need to be performed according to the actual state.Actual maintenances takes into account an important factor of cargo transportation -functioning in conditions of uncertainty and risk [5].
In order to improve the efficiency of maintenance and its evaluation, it is planned to introduce an insurance fund that performs two functions: 1) accumulates payments with different frequency and cost; 2) pays for types of maintenance as necessary.At the same time, taking into account the actual state of transportation infrastructure maintenance under conditions of cargo transportation uncertainty, the time intervals between maintenances (days) and their costs (million rubles) can be considered random variables with known distribution functions up to the values of their parameters.This approach allows for mathematical description of the state of the insurance fund when modeling the efficiency of transportation infrastructure maintenance to use a special type of risk process, which is used in the mathematical theory of risks [6], as well as when evaluating the effectiveness of repair work of complex equipment [7].Using the risk process in describing the insurance fund has been tested in the study of various types of work on the railway track [1,8].

Mathematical description of equipment repairs based on random risk process
Considering that the work investigates three types of repair work, we define the risk process as follows where 0 X is the initial funds of the insurance fund; ( ) Yj t is the total accumulations of payments by type of maintenance, ( ( ) YK t is the total cost for investment maintenance.
When servicing transport infrastructure during its operation, the amount of payments for the year is initially formed ( X , million rubles).Then the annual amount of payments is distributed by type of maintenance ( Here 1 c is the coefficient that takes into account a part of payments for current maintenance; 2 c is the coefficient that takes into account a part of payments for emergency maintenance; 3 c is the coefficient that takes into account a part of payments for investment maintenance; ( ) j X annual payments to the insurance fund by type of maintenance.
The cost of one payment to the insurance fund for the j-th type of maintenance based on (2), are equal ( ) / / , 1, 2,3, where Tg is the number of days in a year; j h are the frequency of payments to the insurance fund (day) for the j-th type of maintenance.The total accumulations of payments to the insurance fund for j-th type performing maintenances both based on (2) and ( 3) and assumptions of replenishment frequency are equal where ( ) j N t is the number of payments to the insurance fund during time t for the j-th type of maintenance.The time intervals between maintenance and the cost of performing them are random variables with known distribution laws.
For a random risk process (1), the time point  is determined when the condition ( ) 0 R t  is fulfilled for the first time.( The moment in time ( 5) characterizes the efficiency of the organization of maintenance in terms of the formation of payments by their types.Therefore, it is proposed to consider a random event ( T    ) as a resource-cost risk and evaluate it by an indicator r  as the probability of this event ( ) where T  is set time (day).
The risk indicator ( 6) is called resource-cost, it evaluates the "Resources-costs" model for maintenance related to the operation of transport infrastructure.If there are no financial resources in the insurance fund, there is a risk of failure to perform the required maintenance.
In practice, the two-factor model is more widely used, when in addition to the probability of a negative event, the financial consequences of the event are considered.In this regard, the paper introduces the concept of financial risk of the following type , million rubles, (7) where R C is the losses from non-performance of repair work, million rubles; r  is the value (6), which characterizes resource-cost risk.In simulation modeling, risks ( 6) and ( 7) are replaced by point estimates where k  is the number of realizations of process (1) for which condition ( 5) is fulfilled, 0 n is the number of created realizations of process (1) by simulation method; 3 The choice of initial data and tasks of computational experiment Based on the developed mathematical software, it created a modeling program to study the efficiency of transportation infrastructure maintenance using the simulation method.It bases the software on the MATLAB programming language, which has several advantages over other software environments designed for performing scientific and engineering calculations [9].The modeling program creates sample values of a special type, which are then processed to obtain the values of performance indicators (8), (9) in the form of risk assessments.
Table 1 shows the distribution laws and their numerical characteristics for the time intervals between types of maintenances and the costs of these maintenances used in this study: , mi mz are the mathematical expectations, v k are the coefficients of variation.With the mathematical expectations selected (Table 1), the average maintenance costs by type for the year are equal / , 1, 2,3 Considering (10) and the values of Table In total, these costs are equal to the annual amount of payments X .Then the share of payments for the types of repairs (2) / , 1, 2,3; If we substitute the values (11) in formula (12), we get the following payment: In the mathematical theory of risks, it is proved that it is necessary to have an average annual excess of income over expenses [6].In our case, the condition must be met In the work [7], it is shown that according to the criterion of the minimum values of resource-cost risk (8), it is best on average to exceed income over expenses (14) at the expense of the annual initial value of the insurance fund 0 X .In this paper, using the created modeling program, we study the impact of the shares and frequency of payments to the insurance fund on the assessment of resource-cost and financial risks base on the recommendations of the work [7].It is proposed to link the frequency of payments ( j h with the mathematical expectations of time intervals between types of work (

The results of a computational experiment
In this study, five options of payments to the insurance fund were modeled: 1) option A, when the shares of payments are equal to (13), the frequency of payments based on ( 15) is equal 2) option B, when the shares of payments are equal to (13), the frequency of payments based on (15)  ); 4) option D, when the shares of payments are equal to (13), the frequency of payments based on ( 15) is equal ) option E, when the payment shares are the same 1 0.333 c = ; 2 0.333 c = ; 3 0.334 c = ; the frequency of payments based on ( 15) is equal ).For each option, two options are considered for the value of the annual initial value of the insurance fund, when these values are equal to 5 and 10 percent relative to the annual of payments The considered options are modeled for four values of the quantity T  (6): 30 days, 90 days, 180 days and 360 days.Losses from non-performance of maintenance (7) are taken in the amount of 250 million rubles.The number of created realizations of the risk process (1) by the simulation method ( 0 n ) is equal to 20000.
Table 2 shows the values of point estimates of resource-cost (8) and financial (9) risks obtained as a result of a computational experiment; V1 and V2 are types of calculations based on the proposed options.
Analysis of the values of table 2 allows us to conclude that in terms of reducing resourcecost and financial risks, the accumulation of the insurance fund should be carried out according to option D, when the share of payments is equal (13), the intervals for replenishing the insurance fund are different, depending on the type of maintenance and initial data.
In option A, the proportionality coefficients (15) are equal to each other, but the estimates of resource-cost and financial risks turned out to be greater than in option D. This is due to the fact that the values T  are multiples of 10, and for the chosen mathematical expectations of the time intervals between the types work ( j mi , Table 1) and equal proportionality coefficients (15), the payment intervals for option A are not multiples of 10.In option D, the proportionality coefficients (15) are most equal to each other, and the intervals between payments are multiples of 10.For option A, the histogram has 6 outliers, and for option D four with a frequency of about 90 days.Especially large emissions are observed with a frequency of 180 days.This is due to the fact that every 180 days in this option, all three payments to the insurance fund coincide.The sample size for option A is 3561, the resource-cost risk estimate (8) is 0.178.For option D, the sample size is 3352, the estimate of the resource-cost risk (8) is 0.168.These data also confirm that option D is more preferable than option A and the other options considered.

Conclusion
In this article research on the efficiency of transportation infrastructure maintenance is conducted and proposed to use an insurance fund.The state of insurance fund is described by a special type of risk process.A program based on the event approach was created to simulate this process.The modeling program creates a sample of times when there are no financial resources available for maintenance.Processing of sample values is carried out on the proposed algorithms for the estimation of resource cost and financial risks, which are performance indicators.As a result of the conducted research, practically important recommendations were obtained that allow determining the shares and frequency of payments to the insurance fund based on the type of maintenance and initial data.

Figures 1 and 2
Figures1 and 2show the histograms of frequencies at 30 intervals of magnitude  (5), when the risk process becomes less than zero for the first time.When 0 14.2 X = million

Table 1 .
Distribution laws and their numerical characteristics.
1, average costs are equal

Table 2 .
The results of calculations of point estimates of risk.