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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

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

Object oriented library for generating and computing stochastic matrix in Markov queueing models

Pecka P.

Archiwum Informatyki Teoretycznej i Stosowanej

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2000

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T. 12, z. 1

37-50

EN

The papers presents a class library which is written in C++ and allows to construct and solve queueing models based on continuous time Markov chains. The C++ application based on this library is time efficient and very flexible. The library contains the objects which describe the work of some high speed networks (e.g. ATM) queueing or control mechanisms (e.g. push-out and threshold space-priority queues, sliding window mechanism). The user can enlarge the library by adding new network elements. The software is especially well suited to analyse transient states and to evaluate various control algorithms that prevent traffic congestion in communications networks.

PL

W artykule przedstawiono bibliotekę klas w języku C++ przeznaczoną do generowania i rozwiązywania macierzy stochastycznych w markowowskich modelach systemów komputerowych. Biblioteka zawiera modele kolejkowe przydatne w ocenie efektywności nowoczesnych, szerokopasmowych sieci komputerowych (np. ATM). Biblioteka udostępnia funkcje ułatwiające dalszą jej rozbudowę o kolejne elementy modeli kolejkowych. Oprogramowanie jest przeznaczone w szczególności do modelowania stanów nieustalonych w systemach komputerowych.