In this paper we propose a method for calculating steady-state probability distributions of the single-channel closed queueing systems with arbitrary distributions of customer generation times and service times. The approach based on the use of fictitious phases and hyperexponential approximations with parameters of the paradoxical and complex type by the method of moments. We defined conditions for the variation coefficients of the gamma distributions and Weibull distributions, for which the best accuracy of calculating the steady-state probabilities is achieved in comparison with the results of simulation modeling.

This article proposes an analysis of the results of the application of hyperexponential approximations with parameters of the paradoxical and complex type for calculating the steady-state probabilities of the G/G/n/m queueing systems with the number of channels n = 1, 2 and 3. The steady-state probabilities are solutions of a system of linear algebraic equations obtained by the method of fictitious phases. Approximation of arbitrary distributions is carried out using the method of moments. We verified the obtained numerical results using potential method and simulation models, constructed by means of GPSS World.

In this paper, we propose a method for studying the reliability of series systems with redundancy and repair facilities. We consider arbitrary distributions of the units' time to failure and exponentially distributed recovery times. The approach based on the use of fictitious phases and hyperexponential approximations of arbitrary distributions by the method of moments. We consider cases of fictitious hyperexponential distributions with paradoxical and complex parameters. We define conditions for the variation coefficients of the gamma distributions and Weibull distributions, for which the best and same accuracy of calculating the steady-state probabilities is achieved in comparison with the results of simulation modeling.

Abstract. We propose a method for determining the characteristics of a single-channel closed queueing system with an exponential distribution of the time generation of service requests and arbitrary distributions of the service times. In order to increase the system capacity, two service modes (the main mode and overload mode), with the service time distribution functions ( ) F x and ( ) % F x respectively, are used. The overload mode starts functioning if at the beginning of service of the next customer the number of customers in the system ( ) ξ t satisfies the condition 2 ( ) . ξ > t h The return to the main mode carried out at the beginning of service of the customer, for whichThe Laplace transforms for the distribution of the number of customers in the system during the busy period and for the distribution function of the length of the busy period are found. The developed algorithm for calculating the stationary characteristics of the system is tested with the help of a simulation model constructed with the assistance of GPSS World tools.

Abstract. We propose a method for determining the probabilistic characteristics of the M/G/1/m queueing system with the random dropping of arrivals and distribution of the service time depending on the queue length. Two sets of service modes, with the service time distribution functions ( ) n F x and ( ) n F x ɶ respectively, are used according to the twothreshold hysteretic strategy. The Laplace transforms for the distribution of the number of customers in the system during the busy period and for the distribution function of the length of the busy period are found. The developed algorithm for calculating the stationary characteristics of the system is tested with the help of a simulation model constructed with the assistance of GPSS World tools.