THE STATES’ FINAL PROBABILITIES ANALYTICAL DESCRIPTION IN AN INCOMPLETELY ACCESSIBLE QUEUING SYSTEM WITH REFUSALS

Abstract

Context. There is a problem of forecasting the efficiency of real queuing systems with refusals in the case of incomplete accessibility of service devices for the input flow of requirements. The solution of problem is necessary to create the possibility of more accurate design and control of such systems operation in real time. Objective. The aim of the research is to obtain an analytical description of the state’s final probabilities in a Markov queuing system with refusals and with incomplete accessibility of service devices for the input flow of requirements that is necessary to forecast the values of the queuing system performance indicators. Method. The probabilities of queuing systems’ states with refusals in the case of incomplete accessibility of service devices for the input flow of requirements are described by Kolmogorov differential equations. In a stationary state, these equations are transformed into a linearly dependent homogeneous system of algebraic equations. The number of equations is determined by the setdegree and for modern queuing and communication systems can be in the thousands, millions and more. Therefore, an attempt to predict the efficiency of a system is faced with the need to write down and numerically solve a countable set of algebraic equations systems that is quite difficult. The key idea of the proposed method for finding an analytical description of final probabilities for a given queuing system was the desire to move from the description of individual states (of 2n amount) to the description of groups of system states (of n+1 number) and to localize the influence of incomplete accessibility of service devices for the input flow of requirements in multiplicative functions of incomplete accessibility. Such functions allow obtaining the required analytical description and assessing the degree of the final probabilities transformation, in comparison with known systems, as well as assessing the forecasted values of the noted queuing system’s efficiency indicators when building a system and choosing the parameters for its controlling. Results. For the first time analytical expressions are obtained for the final probabilities of the queuing system states with refusals and with incomplete accessibility of service devices for the input flow of requirements, which makes it possible to evaluate as well as forecast values of all known system efficiency indicators. Conclusions. The resulting description turned out to be a general case for well-known type of Markov queuing systems with refusals. The results of the numerical experiment testify in favor of correctness the obtained analytical expressions for the final probabilities and in favor of possibility for 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.