Abstract and keywords
Abstract (English):
A model of queueing theory is proposed that describes a queueing system with three parameters, which has important practical applications. The model is based on the continuous time Markov process with a discrete number of states. The model is formalized by a probabilistic space in which the space of elementary events is a set of inconsistent states of the queueing system; and the probabilistic measure is a probability distribution corresponding to a set of elementary events, that is, each elementary event is associated with the probability of the system staying in this state, for each fixed time moment. The model is represented by a system of ordinary differential equations, compiled by methods of queueing theory (Kolmogorov equations). To find the solution of the system of equations, the method of generating functions is used. For the generating function, a partial differential equation is obtained. Finding the generating function completes the construction of a probability space. The latter means that for any random variables and functions defined on the resulting probability space, one can find their probabilistic characteristics. In particular, analytical expressions of the moments (mathematical expectations and variances) of random functions that depend on time are obtained. The peculiarity of finding a solution is that it is obtained not from the probability distribution, but directly from the partial differential equation, which represents a system of ordinary differential equations. For the probability distribution, the solution was found by a combinatorial method, which made it possible to significantly reduce the computations. To apply the formulas in engineering calculations, we consider the stationary case, to which a considerable simplification of the calculations corresponds. A relationship between a system of differential equations and a polynomial distribution known in probability theory is shown. The results are used in the analysis of the reliability of the operation of scalable computing systems; graphical implementation is shown

Queuing theory, Markov process, probability distribution, moments, solution of systems of differential equations, generating functions, method of characteristics
Modern models of queueing theory, because of complexity, are rst formulated descriptively and then formalized. Most often, stationary probabilistic characteristics are sought [1].If the model is formalized by differential equations, then the probabilistic characteristics can be written either analytically or approximately. Analytical view is characteristics represented as a function of time (t < ∞), or stationary (t → ∞).In models formalized by systems of differential equations, the unknown functions of time are usually represented by the probability distribution. The analytical solution of such systems is, as a rule, difcult to see. This leads to the fact that the use of standard methods that require a large amount of computation, in the process of their implementation, lose their connection (exists independently) with the descriptive formulation of the model. Thus, the solution found is perceived as a “long set of little informative formulas,” that is, such an analytical solution does not justify the informative hopes usually associated with analytic formulas. Then the properties are used that the problem poses, which allow us to use particular approaches to nd the solution [2, 3].The aim of this paper is to construct a probabilistic space describing a queueing system with three parameters. To achieve the goal, it is necessary to obtain a differential equation that allows to calculate the necessary probabilistic characteristics. In addition, to show the applicability of the results

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