THE ANALYTICAL DESCRIPTION OF FINAL PROBABILITIES FOR STATES OF QUEUING SYSTEMS WITH INPUT FLOW OF GROUPS OF REQUIREMENTS

Abstract

Context. The management of many economic and other “service” systems of random flows of “requirements” is based on the prediction of their efficiency, based on an estimate of the system states probability distribution. In a number of important practical cases, the input flow may have random composition groups of requirements, which determined the applicability of linear algebra numerical methods for searching probabilities, and also made it difficult to build queuing systems that are effective in a range of conditions and made it impossible to obtain probability estimates for systems with an infinite number of places to wait for service. The objects of the study are Markov models of three types of queuing systems: with refusals, with a limited and with an unlimited number of places to wait in the conditions of the input flow of a random composition groups of requirements. Objective. The goal of the work is to obtain an analytical description of the final state probabilities which are necessary to predict the values of efficiency indicators for three types of Markov multichannel queuing systems: with refusals, with a limited and with an unlimited number of places to wait in the conditions of the input flow of random composition groups of requirements. Method. In the general case, the probabilities of states in queuing systems with input flow random groups of requirements are described by Kolmogorov differential equations. The Kolmogorov equations, in the stationary state of the queuing system, are transformed into a linearly dependent homogeneous system of algebraic equations. The final probabilities of the states of a queuing system can be found by numerically solving a system of equations using methods well known in linear algebra: complete exclusion, the inverse matrix, and the matrix method of Ramaswami [3], [38], which takes into account the repeating block structure of the system of equations matrix. The infinite number of unpredictable combinations for the set of numerical values of the considered queuing systems parameters makes it difficult to control the operation of such systems and to build systems that are effective in a range of conditions. In queuing systems with an unlimited number of places to wait, the number of equations becomes infinite, and numerical methods become unsuitable for final probabilities searching and for solving problems of analysis, synthesis and control of queuing systems. Analytical expressions for the final probabilities of queuing systems are obtained by equivalent transformations of homogeneous systems of algebraic equations in the general case of each type of queuing system mentioned above. Results. The obtained analytical expressions for the final probabilities of the queuing systems states for three noted system types are not previously known and therefore required verification of their correctness. Such a check was performed by the way of degenerate the flow of random groups of requirements in the input of the system to the simplest flow of requirements. As a result of verification, analytical expressions for the final probabilities of the considered systems states were automatically transformed into the corresponding well-known models of queuing systems with the simplest input flow of requirements. This effect allows us to consider the well-known models of queuing systems of the simplest input requirements flow – to be a particular case of the obtained models of queuing systems with an input flow of groups of requirements. Conclusions. To further verify the correctness of the results and to assess the degree of influence of requirements random number in groups of input flow onto the system efficiency, a numerical example is given for the critical conditions of a constant intensity of requirements flow equal to the total performance of the system’s service channels. In this case, only the average number of requirements in groups changed. The results of the numerical experiment testify in favor of the correctness of the obtained analytical expressions for the final probabilities and in favor of the possibility of their practical application in real queuing systems when solving problems of forecasting efficiency, as well as analyzing and synthesizing the parameters of real queuing systems.