The States Final Probabilities Analytical Description in Queuing System With an Entrance Flow of Requirements Groups, With Waiting and Leaving the Queue

Abstract

Context. The problem of predicting the efficiency of real queuing systems in the event of a possible arrival of requirements groups and leaving of "impatient" requirements from the queue. The aim of the study was to model the operation of such systems to create opportunities to control their 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 an input flow of requirements groups, with individual service of requirements, with a limited number of waiting places and with individual leaving of "impatient" requirements from the queue that is necessary to predict the values of the queuing system performance indicators.Method. The probabilities of queuing systems states with an input flow of requirements groups with a random composition and with leaving of "impatient" requirements from the queue are described by the Kolmogorov differential equations. In a stationary state, these equations are transformed into a linearly dependent homogeneous system of algebraic equations. The structure of the equations depends on the numerical values of the input flow requirements group's parameters and the controlled service system. 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 the final probabilities for the specified queuing system was the desire to localize the influence of requirements groups in the input flow on the operation of the queuing system in multiplicative non-ordinary functions. 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 predicted values of the noted queuing system efficiency indicators when choosing the parameters for controlling its operation.Results. For the first time analytical expressions are obtained for the final probabilities of the queuing system states with an input flow of random composition requirements groups, with a limited number of waiting places, with individual service and leaving "impatient" requirements from the queue, which makes it possible to evaluate all known indicators of the system's performance.Conclusions. The resulting description turned out to be a general case for well-known types of Markov queuing systems with non-ordinary and with the simplest input flow of requirements. 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.