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Content available Markov reliability models of series systems with redundancy and repair facilities
Zhernovyi Yuriy, Kopytko Bohdan
Journal of Applied Mathematics and Computational Mechanics
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2021
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Vol. 20, nr 3
89--96
EN
In this paper, we use Markov models for studying the reliability of series systems with redundancy and repair facilities. We suppose that the units’ time to failure and recovery times are exponentially distributed. We consider the cases when 1≤ c ≤ m and m + 1 ≤ c ≤ m + n, for the system of n operating units, m unloaded redundant units and c repair facilities. Using the exponential distributions properties, we obtain stationary reliability indices of the series systems: steady-state probabilities, a stationary availability coefficient, mean time to failure, mean time between failures and mean downtime.
2
Content available Formulas for average transition times between states of the Markov birth-death process
Zhernovyi Yuriy, Kopytko Bohdan
Journal of Applied Mathematics and Computational Mechanics
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2021
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Vol. 20, nr 4
99--110
EN
In this paper, we consider Markov birth-death processes with constant intensities of transitions between neighboring states that have an ergodic property. Using the exponential distributions properties, we obtain formulas for the mean time of transition from the state i to the state j and transitions back, from the state j to the state i. We found expressions for the mean time spent outside the given state i, the mean time spent in the group of states (0,...,i-1) to the left from state i, and the mean time spent in the group of states (i+1,i+2,...) to the right. We derive the formulas for some special cases of the Markov birth-death processes, namely, for the Erlang loss system, the queueing systems with finite and with infinite waiting room and the reliability model for a recoverable system.
3
Content available Reliability assessment of a series system with redundancy and repair facilities
Zhernovyi Yuriy, Kopytko Bohdan
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 3
123--131
EN
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.
4
Content available Calculating steady-state probabilities of single-channel closed queueing systems using hyperexponential approximation
Zhernovyi Yuriy, Kopytko Bohdan
Journal of Applied Mathematics and Computational Mechanics
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2020
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Vol. 19, nr 1
113--120
EN
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.
5
Content available Calculating steady-state probabilities of single-channel queueing systems with changes of service times depending on the queue length
Zhernovyi Yuriy, Kopytko Bohdan
Journal of Applied Mathematics and Computational Mechanics
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2019
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Vol. 18, nr 3
107--117
EN
In this paper, we propose a method for calculating steady-state probabilities of the G/G/1/m and M/G/1/m queueing systems with service times changes depending of the number of customers in the system. The method is based on the use of fictitious phases and hyperexponential approximations with parameters of the paradoxical and complex type. A change in the service mode can only occur at the moment the service is started. We verified the obtained numerical results using the potential method and simulation models, constructed by means of GPSS World.
6
Content available Calculating steady-state probabilities of queueing systems using hyperexponential approximation
Zhernovyi Yuriy, Kopytko Bohdan
Journal of Applied Mathematics and Computational Mechanics
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2019
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Vol. 18, nr 2
111--122
EN
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.
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