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Study of Ergodic Characteristics of the System M | G | n | ∞ in Which the Service Time is a Mixture of Exponential Distributions

Student: Svetlana Polozova

Supervisor: Yulia Grishunina

Faculty: HSE Tikhonov Moscow Institute of Electronics and Mathematics (MIEM HSE)

Educational Programme: Applied Mathematics (Bachelor)

Year of Graduation: 2018

Abstract The object of the study in the work is a non-Markov queuing system M | G | n | ∞, with a random number of application types and service time in the form of a mixture of exponential distributions. In the course of writing the work, an N-dimensional Markov process was constructed that describes the functioning of the SMO that is being studied, which provides access to a system with an random number N of serving devices in the system. We prove the assertion about reduction of an infinite system of equations for the limiting distribution to a finite system. A stationary distribution is obtained for the non-Markov system under study A comparative analysis of the results is performed with the corresponding values of the stationary distribution in the Markov case, the coefficients at which the Markov model approximates the model studied in the paper. The main goal of the study was to calculate the stationary distribution for a non-Markov system with different applications. Such systems can be used as mathematical models of real queuing systems, and the results obtained will be closer to real ones than in Markov models. In addition, for systems of this kind there are no standard formulas, and many of the existing varieties of such systems have not been studied. All this causes the novelty and relevance of the results obtained.

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