Optimization of queuing systems under significant loads

The article deals with analytical models of Markov Queuing systems with service failures, incoming requests and requirements. The systems are analyzed in the conflict situations, for example, under significant loads, when the input flow intensity is high relative to the service intensity, which is important for extreme situations, both in technical applications, Internet applications, and social ones. There occurs a problem of optimization - minimization of the number of channels, provided the Queuing system has a guaranteed throughput. There is considered the approach to solving the optimization problem, when the relative system throughput is maximized while minimizing the number of service channels. Given the fact that the analytical formulas of Markov Queuing systems contain factorials, the analytical analysis of systems encounters the computational limitations. In the conducted research, in order to resolve computational difficulties it was decided to apply the approximation of the probabilities of the system states using the Laplace probability integral. Its use is justified precisely at high system load rates and a large number of service channels. There are described the features of applying the Laplace integral in conjunction with the numerical optimization for a conditional extremum. There is given the method of determining the number of service channels, when the probability of denial of service is minimized, respectively, maximizing the relative throughput of the system. There is given a graphical interpretation of the proposed method for optimizing Queuing systems with failures at the significant load. It is shown that during the search for the optimum there is a transition process in which there take place the significant changes in the system parameters: the intensity of the input flow and the intensity of service.