Problem Relevance. Coal-mining industry development is defined as the one changing to deeper horizons with more sophisticated mining conditions, which can cause the level of labor safety decrease. The problem of labour protection in mining makes high demands to the safety and efficiency of mine systems functioning, especially, to a most important of them — mine ventilation system. The violations of mine ventilation systems’ normal functioning can cause terrible economical and social aftereffects.
        Ventilation networks in mines consist of a whole set of numerous underground openings, which differ from each other with various parameters influencing the aerodynamics of these nets. These parameters are constantly changing, therefore the aerodynamics of the nets is changing as well. This process is defined as a set of various geological, manufacturing and technical factors, the majority of which are stochastic (accidental). These factors cause stochastic dynamics of the outline and parameters of mine ventilation systems, which isn’t taken into account now while developing a mine field. As a result, there appear violations in mine ventilation systems, i.e. a system breakdown.

        General formulation The purpose of a mine ventilation system is to supply the necessary amount of air with definite fan pressure to a mine. Hence the function of mine ventilation systems can be defined as a continuous air supply, needed to maintain normative (i.e. meeting the requirements of safety regulations) conditions of a mine atmosphere. That’s why a reliable mine ventilation system lies in the ability to preserve the air distribution in the net of mine openings while a mine is functioning.

        It means that while a mine ventilation system is functioning any moment t in any mine opening the actual air consumption Q_i(t) should be in the range of legitimate value. The lower (left) value of this range is equal to the maximum of the next two values: Qi(t) air consumption, required by gas, heat and dust factors, and also by the number of workers in the mine; Q_i(s) — air consumption, required by the minimum allowable speed, i.e.

        Therefore, the sign of a mine ventilation system being reliable is carrying out of the following inequality any time t:


n_b – the number of a mine ventilation system’s branches.

        The output effect of a mine ventilation system is the degree of correspondence between factual and required air consumption. The actual airflow overrun means the ventilation system failure. The greater the deviation of an actual air flow rate Q from the boundaries of the interval of its admissible values, the less efficient functioning of mine ventilation systems is. The quantity of the air supplied to the mine depends on the following processes: the wastewater treatment works, roadheading, the repayment roadway, replacement and addition of the head fan, aerodynamic aging of roadway, changing needs in the air. Depending on these processes, between the moments of time t_j-1 and t_j may occur some events. Taking into concideration the complex process of ventilation systems in mines, limit them in this work and consider the processes occurring in the sewage treatment works.

        Random changes in time velocity face advance can be both gradual (as a result of continuous variation of the actual values of process parameters from their design value) and sudden (due to equipment failure or geological disturbances). Randomness velocity face advance due to the gradual, random changes of mine ventilation systems parameters, as changes in resistance due to changes in the workings of their lengths in the process of elaboration of the mine field. In addition, gradual changes in the parameters of mine ventilation system are the change in resistance as a result of the workings of their aerodynamic aging (clamping of the workings, worsening over time tightness ventsooruzheny). Sudden change the WAN is a variation of the resistance elements of the network due to their failure (the collapse of excavations, destruction of ventilation constructions). Gradual changes in the mine ventilation system parameters of random shall be called "aging" of its elements. Sudden changes in the same, i.e. "fracture" in his work shall not be considered.

         Solution of the problem and the results of the research work. To develop a mine ventilation system model was selected one of the methods of mathematical modeling of technical systems’ reliability — simulation, the subject of which is the very process of functioning of the mine ventilation system, i.e. system behavior in its development. The advantage of simulation method is the high degree of adequacy of the model and a comprehensive analysis of the system behavior, and the weakness is the complexity of computing.

         The solution to problems controlling the air supply on a mathematical model of the ventilation system allows a large number of experiments in a relatively short period of time, and analyzes the state of the ventilation system when you change the settings, and the range of variation of parameters is almost unlimited (i.e., the mathematical model can simulate virtually any situation). Gradual changes in time of resistance due to its aerodynamic aging is a random process R(t). The random process is its distribution law (in the study of many of the technical facilities it was suggested that the intensity of the flow failures and restorations is constant, i.e. random flows of failures have the define nature) the mathematical expectation m(t), the variance σ(t)² and the correlation function K(t). To obtain estimates of statistical characteristics it is necessary to analyze the data obtained through experimental studies. Experimental studies lie in the fact that similar in mining geology and mining engineering factors of longwalls are distinguished group of mines, in which measurements are taken. The time variation of resistance of a definitely observed object represents an individual realization of the random process of aging R(t), , which is the characteristic of mine ventilation system’ items similar to the observed object subtype, then the measurement values for each of the realizations of the random workings of the aging process.

         Experimental studies in the mines are quite complex and require more material and technical costs, so the work will use statistical data obtained previously for similar systems. On the basis of experimental data, averaged for each sample, i.e. the realizations, we’ll define the sample mean mR(tj) and sample variance DR(tj). The correlation function of the random process of aging R(t) because of its unsteadiness (as even its expectation mR(t)≠ const is a function of two variables, KR(t', t"). Evaluation of the correlation function of this type — is quite difficult and time-consuming task . However, the original process R(t) can be converted to a stationary form, considering the so-called normalized aging process R0(t). In this case the random variables R_i(t_j) at each time counting t_j are transformed as follows:

        The obtained sequence of random variables is stationary, and hence the normalized correlation function of the aging process R0(t) is a function of one variable — the time interval between readings τ = (t', t") , т.е. K_R(t', t") = K₀(τ ), and its score is determined by simply averaging over realizations. However, for those moments of time t_j, the number of observed realizations n_j of process R(t) is small, estimation of the correlation function obtained by averaging over the realizations, will have a low accuracy. Assuming ergodicity of the normalized random process of aging R₀(t), can obtain a more accurate estimate, using the averaging over the realizations of the time average of any one of ith of the realizations:

        The legitimacy of the ergodicity assumption of the process R0(t) due to zero when τ bigest then infinity is its correlation function. The final evaluation of the correlation function of KR(t', t") (t" = t'+ τ ) of the aging process source of the R(t) has the form:

        The type of interpolating models m_R(t), σR(t) and K₀(j), which approximate the obtained discrete times t_j of statistical characteristics’ evaluation with the help of analytic functions, selected on the basis of the following. Random aging workings R(t) «noisier» flow of periodic preventive maintenance (retimbering), so the time change of the statistical characteristics of the process R(t) should also be periodic. Therefore, in the expressions for mR(t), σR(t) and K_j, there must be present trigonometric functions.

        Based on the foregoing, the functions that approximate the statistical characteristics of the aging process workings R(t), chosen in the form:

        Included in the expressions (6) — (8) the coefficients A, B, C and D are determined by the method of least squares. This method is one of the most common methods of statistical processing of experimental data pertaining to different functional dependence of physical quantities from each other. Including, it is applicable to linear dependence and to obtain reliable estimates of parameters. Selected interpolation model (6) — (8) satisfactorily describe the experimental evaluation of the statistical characteristics of m_R(t), σR(t) and K₀(τj ).

         In addition to estimates of the expectation, standard deviation and correlation function of a random model of the aging process requires knowledge of the workings of its law distribution. To determine the form of the distribution of this random process for the various sections of his time t_j we will construct a histogram . Then programmatically we will assess against the densities of the nine most common laws of distribution by using the criterion of the Kolmogorov - Smirnov significance level a = 0,05. For all classification groups of workings aging process R(t) has a lognormal distribution with the density distribution of the form:

         Parameters of lognormal distribution of ? and ? are determined by the estimated statistical characteristics of m_R and σR as follows:

         A random process of aerodynamic aging of workings R(t) and its statistical characteristics (6) — (11) reflect the gradual change in resistance workings. Based on these estimates, the statistical characteristics of aging mines, we can simulate the random process of aging generation, using some of the methods of modeling stationary normal random processes. Recently, very efficient modeling algorithms for stationary normal random processes have been found. Let’s consider one of the simulation algorithms of stationary normal random processes, which is based on a linear transformation of a stationary sequence x(t_j) of the independent random numbers (discrete white noise) in the sequence y(t_j), which has given the correlation coefficient K(τ). The operator of a linear transformation is a recursive relation, which for a given type of correlation coefficient K(τ), the values of m(t_j), the variance σ(t_j), as well as the values of the process at previous times y(t_ji-k) is defined " aged" meaning (one of the possible) of the process the next time y(t_j). This operator is a linear transformation can be written either in the form of moving summation with a weight сk= c[k]

         Conclusion: Within the framework of a deterministic model of the system adopted in the traditional methods of ventilation calculation, the stochastic dynamics of the functioning of mine ventilation systems cannot adequately be taken into account, which causes system’s failure. So it is necessary to take into account changes in the aerodynamic parameters of the workings within the stochastic model. One of the main steps of the workings modeling of the aging process is to obtain its statistical estimates.

         References.
1. Ushakov, VK, "The reliability and efficiency of mine ventilation systems", 2003.
2. Emelin PV "Theoretical problems solving regulation of air distribution in the mine ventilation systems", 2005.
3. Kelton B., Lowe L., "Simulation", Moscow, 2004.