# Reducing the Stochastic Coal Output Model Using the Convolution of Probability Density Functions
###### Abstract
In [10], stochastic methods for analyzing the production process carried out in longwall faces were presented. Analytical modeling is also used for issues related to natural hazards [11; 12], the optimization of technological flows [13; 14; 15; 16; 17], extraction volume [18], mine planning [19; 20; 21], and even in the design of computer systems used to improve the efficiency of mine rescue systems. Random variables that occur in stochastic models may be subject to probabilistic relationships. One of the most common probabilistic relationships for random variables is their sum (e.g., the total process time as the sum of its individual fragments, the total employment as the sum of staff, the daily output from a longwall face as the sum of the extraction obtained in individual work shifts, etc.). In the conducted studies [22; 23], the usefulness of this type of models was noted. Therefore, in the subsequent part of this paper, the authors put forward the hypothesis that it is possible to use the properties of the convolution of a random variable function to reduce the stochastic model, in particular to model the processes carried out in the longwall faces of hard coal mines. Recent decades have seen growing interest in the use of stochastic approaches in various areas of mining engineering. [PERSON] [13] proposed probabilistic models tailored to the production systems in underground coal mines, highlighting the necessity of incorporating real-time variability into system modeling. [PERSON] et al. [14] introduced stochastic Petri nets to model disaster chains triggered by mining subsidence, demonstrating the flexibility of stochastic structures in representing complex interdependencies. Additional research has focused on optimization of the technological flows and production scheduling under uncertain conditions. For example, [PERSON] et al. [15] and [PERSON] et al. [16] showed how geological and operational uncertainties affect production efficiency and reliability, integrating stochastic parameters into the optimization algorithms. This approach also appears in studies that apply Monte Carlo simulations to rockburst predictions [11] and stochastic fracture networks to risk modeling in waste repositories [8]. Beyond process-level modeling, stochastic methods have also been used in mine planning and risk assessments. [PERSON] et al. [20] incorporated the uncertainty in coal quality and reserves into mine scheduling, while [PERSON] et al. [21] applied stochastic tools to assessing the social and environmental risks in surface mining. Particularly notable is the application of stochastic modeling to time-based production analyses--such as shift-wise or daily output modeling--where convolution of probability density functions offers a practical way to model the cumulative variability. [PERSON] et al. [22; 23] emphasized how process fragments (e.g., the time durations of sub-activities) can be modeled as random variables and their summation described analytically via convolution. This approach avoids unnecessary simulations while preserving the model's fidelity and transparency. This modeling strategy aligns with the central limit theorem, which justifies the approximation of sums of independent random variables using normal distributions under certain assumptions. Depending on the process characteristics, other distributions--such as chi-square and gamma--are also frequently used to reflect specific empirical observations better [24; 25]. These developments confirm that accounting for stochastic dependencies is indispensable in modern mining research. This paper contributes to the field by proposing a method for stochastic model reduction using the convolution properties of probability density functions, with particular attention to longwall mining operations. This simplification enhances both the analytical clarity and practical applicability of the models in mining practice.