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1
Content available Monte Carlo method for reliability of parallel system with dependent failures of component
100%
Grabski F.
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2016
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tom Vol. 7, No. 1
67--72
EN
A problem of parallel system reliability with dependent failures of components is presented in the paper. It is assumed that lifetimes of components are independent random variable having Weibull distribution. We take under consideration a parallel (in reliability meaning) system consisting of n independent at the beginning of work and identical components. We assume that a load of the working system affects on the reliability of its components and the load of the system is distributed on all working components. Therefore, a failure rate of each component is changeable during run of the system and depends on a number of working elements at this point in time. As a model of the system failures we construct a stochastic process which value at the moment t denotes the number of working components. Generally it is neither Markov nor semi-Markov process. To assess the reliability characteristics of the system we simulate this stochastic process using the Monte-Carlo method and we calculate values of nonparametric kernel density and reliability functions estimators.
2
Content available The renewal process generated by return times of semi-Markov process in reliability models
100%
Grabski F.
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tom No. 43
365--380
EN
The renewal process generated by the return times of semi-Markov process to a given state is considered in the paper. The return time to a state j and also a first passage time from a given state i to the state j of semi-Markov process are basic concepts that are used to determine this process. The systems of equations for distributions, expectations and second moments of these random variables are presented. Theorem concerning the asymptotic distribution of the considered renewal process is presented in this article. Moreover an illustrative example from the reliability theory is presented in the paper.
PL
W artykule jest rozważany proces odnowy generowany przez czasy powrotu procesu semi-Markowa (SM) do danego stanu. Czas powrotu do stanu j, a także czas pierwszego przejście od danego stanu i do stanu j semi-Markowa procesu są podstawowymi pojęciami, które są stosowane do określenia tego procesu. W pracy zostały przedstawione układy równań dla transformat rozkładów, wartości oczekiwanych i drugich momentów tych zmiennych losowych. Twierdzenie dotyczące asymptotycznego rozkładu badanego procesu odnowy jest kluczowym elementem pracy. Ponadto został przestawiony przykład z teorii niezawodności ilustrujący rozważane problemy.
3
Content available Nonhomogeneous stochastic processes connected to Poisson process
100%
Grabski F.
Zeszyty Naukowe Akademii Marynarki Wojennej
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2018
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tom R. 59 nr 2 (213)
5--15
EN
Some generalizations of the Poisson process and their properties are presented in the paper. The non-homogeneous Poisson process allows to construct a probabilistic model describing the different kinds of accidents number. The nonhomogeneous compound Poisson process enables to describe mathematically the various types of accidents consequences. Theoretical results give possibility to anticipate the accidents number and their consequences.
PL
W artykule przedstawiono wybrane uogólnienia procesu Poissona i ich własności. Skupiono się na dwóch uogólnieniach — niejednorodnym procesie Poissona i niejednorodnym złożonym procesie Poissona. Niejednorodny proces Poissona pozwala na skonstruowanie modelu probabi-listycznego opisującego liczbę różnych rodzajów wypadków. Niejednorodny złożony proces Poissona pozwala matematycznie opisywać konsekwencje tych wypadków. Przedstawione tu wyniki teoretyczne dają możliwość przewidywania liczby wypadków i ich konsekwencji.
4
Content available Semi-Markov processes: application in system reliability and maintenance
100%
Grabski F.
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2014
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tom Vol. 5, No. 1
57--62
EN
The author’s monograph “Semi-Markov Processes: Application in System Reliability and Maintenance” which will be published by Elsevier in 2014 is presented. The paper is composed of an introduction, the monograph contents, conclusions and the references the monograph contents is based on.
5
Content available Semi-Markov model of multi-modal transport operation
100%
Grabski F.
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2017
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tom Vol. 24, No. 4
47--54
EN
Multi-modal transport means the transport of the objects through at least two different carriers of any combination of simple tasks of transport carriers (by truck, by train, by ship or by plane). A Semi-Markov (SM) model of multi-modal transport operation is presented in the article. The SM process is defined by the renewal kernel of that one. In our model, time to failure of the operation is represented by a random variable that denotes the first passage time from the given state to the subset of states. The duration of one operation cycle is a random variable representing the return time to the initial state. The appropriate theorems of the Semi-Markov processes theory allow us to calculate characteristics and parameters of the transport operation model. The article presents the example of the transport operation final part of container with cargo from Warsaw to Stockholm, where from Warsaw to Gdynia, the container is transported by lorry, from Gdynia to Karlscorona by ferry and from Karlscorona to Stockholm by truck.
6
Content available remote Reliability electrical power system of hospital as cold standby system
100%
Grabski F.
|
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tom No. 1 (37)
153--167
EN
The probabilistic model of a hospital electrical power system consisting of mains, an emergency power system and the automatic transfer switch with the generator starter are discussed in this paper. The reliability model is semi-Markov process describing two different units renewable cold standby system and switch. The embedded Semi-Markov processes concept is applied for description of the system evolution. Time to failure of the system is represented by a random variable denoting the first passage time of the process from the given state to the subset of states. The appropriate theorems of the Semi-Markov processes theory allow us to evaluate the reliability function and some reliability characteristics.
PL
W pracy został przedstawiony model niezawodnościowy sytemu elektroenergetycznego szpitala złożony z podsystemu zasilania sieciowego, podsystemu zasilania awaryjnego oraz automatycznego przełącznika. Niezawodnościowym modelem funkcjonowania systemu jest proces semi-markowski. Model ten jest modyfikacją modelu niezawodności opisującego funkcjonowanie sytemu z rezerwą zimną złożonego z dwóch różnych podsystemów i przełącznika. Do konstrukcji modelu został wykorzystany tak zwany włożony proces semi-markowski. Czas zdatności systemu jest reprezentowany przez czas pierwszego przejścia procesu do określonego podzbioru stanów.
7
Content available Reliability and maintainability characteristics in semi-Markov models
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Grabski F.
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2016
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tom Vol. 7, No. 1
79--86
EN
The characteristics of semi-Markov process we can translate on the reliability characteristics in the semi-Markov reliability model. The cumulative distribution functions of the first passage time from the given states to subset of states, expected values and second moments corresponding to them which are considered in this paper allow to define reliability function of the system. The equations for many reliability characteristics and parameters are here presented.
8
Semi-markowski proces decyzyjny jako model optymalizacji procesu eksploatacji statku
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Grabski F.
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2011
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tom nr 6
PL
W pracy zostało przedstawione zagadnienie optymalizacji procesu eksploatacji statku. Do matematycznego opisu i rozwiązania problemu została wykorzystana teoria semimarkowskich procesów decyzyjnych. Wykorzystano model decyzyjnego procesu decyzyjnego w nieskończonym przedziale czasu.
EN
A problem of optimization of a ship's operation is discussed in the paper. To describe and solve this problem, a semi-Markov decision processes theory is applied. The infinite duration semi-Markov decision process as a model of the sea transport operation is constructed. An algorithm which allows to compute the optimal strategy of the operation in this paper is presented.
9
Content available Semi-Markov model of system damage process
100%
Grabski F.
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tom Vol. 2, No. 1
101--108
EN
The reliability characteristics and parameters of multi-state systems modelled by the finite states regress characteristics and parameters of multi-state systems modelled by the finite states regress semi-Markov processes are investigated in the paper. Presented here models deal with un-repairable systems. The essential concepts of discrete states and continuous time semi-Markov process theory deliver. Mathematical apparatus for models constructions and analysis. Multi-state reliability functions and corresponding expectations, second moments and standard deviations are calculated for the presented systems.
10
Content available Random Failure Rate
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Grabski F.
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2012
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tom Vol. 3, No. 2
209--216
EN
The reliability function defined by a failure rate which is a stochastic process with nonnegative and right continuous trajectories is presented in this paper. The reliability function with an at most countable state space semi-Markov failure rate process is investigated. A theorem concerning of equations for a conditional reliability function with a semi-Markov process as a failure rate is presented in this paper. The solutions of the proper renewal equations allow getting the reliability functions for the finite space semi-Markov random walk, for the Poisson process and for the Furry-Yule process as a failure rate.
11
Content available The random failure rate
100%
Grabski F.
|
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tom Vol. 1
143--149
EN
A failure rate of the object is assumed to be a stochastic process with nonnegative, right continuous trajectories. A reliability function is defined as an expectation of a function of a random failure rate process. The properties and examples of the reliability function with the random failure rate are presented in the paper. A semi-Markov process as the random failure rate is considered in this paper.
12
Content available Semi-Markov reliability model of two different units cold standby system
100%
Grabski F.
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2017
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tom R. 58 nr 4 (211)
45--60
EN
A semi-Markov stochastic process is used for solving in a reliability problem in the paper. The problem concerns of two different component cold standby system and a switch. To obtain the reliability characteristic and parameters of the system we construct so called an embedded semi-Markov process in the process describing operation process of the system. In the model the conditional time to failure of the system is represented by a random variable denoting the first passage time from the given state to the specified subset of states. We apply theorems of the Semi-Markov processes theory concerning the conditional reliability functions to calculate the reliability function and mean time to failure of the system. Often an exact reliability function of the system by using Laplace transform is difficult to calculate, frequently impossible. The semi-Markov processes perturbation theory, allows to obtain an approximate reliability function of the system in that case.
PL
Do rozwiązania problemu z zakresu teorii niezawodności został zastosowany proces semi-Markowa. Problem dotyczy tak zwanego systemu z rezerwą zimną, który jest złożony z dwóch różnych podsystemów i przełącznika. Aby uzyskać charakterystyki i parametry niezawodności tego systemu, jako model funkcjonowania systemu konstruujemy proces semi-Markowa — tak zwany proces włożony w inny proces stochastyczny. W naszym modelu czas zdatności systemu jest reprezentowany przez zmienną losową oznaczającą czas pierwszego przejścia z danego stanu do określonego podzbioru stanów. W celu obliczenia funkcji niezawodności i średniego czasu do awarii systemu stosujemy twierdzenia teorii procesów semi-markowskich dotyczące warunkowej funkcji niezawodności. Najczęściej dokładna funkcja niezawodności systemu przy zastosowaniu transformaty Laplace’a jest trudna do wyliczenia. W takim przypadku teoria zaburzonych procesów semi-markowskich pozwala otrzymać przybliżoną funkcję niezawodności systemu.
13
Content available Nonhomogenous Poisson process application to modelling accidents number at Baltic Sea waters and ports
100%
Grabski F.
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2017
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tom Vol. 8, No. 1
39--46
EN
The stochastic processes theory provides concepts and theorems that allow to build probabilistic models concerning incidents or (and) accidents. Counting processes are applied for modelling accidents number in Baltic Sea region in the given time interval. A crucial role in construction of the models plays a Poisson process and its extensions especially a nonhomogeneous Poisson process. The models of accidents number in the sea and seaports are here presented. Moreover some procedures of the model parameters identification are presented in the paper. Estimation of model parameters was made based on data from reports of HELCOM (2014) and Interreg project Baltic LINes (2016-2019).
14
Content available Stochastyczny model bezpieczeństwa obiektu w procesie eksploatacji
100%
Grabski F.
|
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tom nr 1
89-102
PL
Teoria procesów stochastycznych dostarcza pojęcia i twierdzenia umożliwiające matematyczny opis i analizę różnych aspektów funkcjonowania systemów, a w tym aspektu bezpieczeństwa. W pracy jest przedstawiony przykład modelu procesu eksploatacji obiektu w aspekcie bezpieczeństwa. Modelem jest proces semimarkowski o skończonym zbiorze stanów. Teoria procesów semi-markowskich pozwala określić parametry i charakterystyki bezpieczeństwa.
EN
To describe the safety evolution of the system, we constructed a Semi-Markov process by defining the states and the renewal kernel of that one. In our model, the time of the safety system operation is represented by a random variable that denotes the first passage of time from the given state to the subset of states. Appropriate theorems from the Semi-Markov processes theory allow us to calculate the safety function and the mean time of the safety operation.
15
Content available Application of perturbed semi-Markov processes in reliability
100%
Grabski F.
Journal of Polish Safety and Reliability Association
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2013
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tom Vol. 4, No. 2
179--188
EN
The paper is concerned with the application of perturbed semi-Markov (SM) processes in reliability problems. There are two kinds of perturbed SM processes presented in the paper. First of them was defined by Shpak and the second one was introduced by Pavlov and Ushakov. Shpak’s concept of perturbed SM is applied for calculating the approximate reliability function of many tasks operation process and Pavlov and Ushakov concept of that one is used to obtain the approximate reliability function of a repairable cold standby system with a switch.
16
Content available Decision problem for infinite duration semi-Markov process
100%
Grabski F.
|
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tom Vol. 5, No. 2
33--40
EN
In the paper there are presented basic concepts and some results of the theory of semi-Markov decision processes. The optimization problem for the infinite duration SM process is connsider in the paper. The Howard algoritm which enables to find the optimal stationary strategy is also discussed here. The algorithm is applied in a decision problem concerning the two components renewable series system is. It is also shown that this algorithm is equivalent to the some linear programing problem.
17
Content available Applications of bootstrap and resampling methods in empirical Bayes estimation of reliability parameters
63%
Grabski F. , Załęska-Fornal A.
Journal of Polish Safety and Reliability Association
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2008
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tom Vol. 1
117--121
EN
Bootstrap and resampling methods are the computer methods used in applied statistics. It is a type of Monte Carlo method based on observed data. Bradley Efron described it in 1979 and he has written a lot about the method and its generalizations since then. Here we apply these methods in an empirical Bayes estimation using bootstrap or resampling copies of the data to obtain an empirical prior distribution.
18
Content available remote Wybrane problemy diagnostyki z wykorzystaniem metod bayesowskich
63%
Jaźwiński J. , Grabski F.
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2004
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tom Vol. 53, nr 5
19-31
PL
W latach 1996-1999 realizowany był grant pod kierownictwem profesora Franciszka Grabskiego nt. "Estymacja bayesowska parametrów i charakterystyk niezawodności na przykładzie systemu transportu lotniczego i morskiego" (nr OT 00A03412). Niektóre wyniki uzyskane w grancie zostały opublikowane w książce: "Metody bayesowskie w niezawodności i diagnostyce", WKiŁ, Warszawa 2001. Niniejsza publikacja ujmuje niektóre problemy z tej książki. W pracy sformułowano pojęcia takie jak: stan obiektu, stan awaryjny, czas ekspozycji, współczynnik pokrycia. Pojęcia te charakteryzują obiekt podlegający diagnozie. Następnie rozważono problemy rozpoznania i ich charakterystyki. Sformułowano pojęcie klasy. Rozważono diagnostykę logiczną i wskazano na zbieżność diagnoz z pojęciem klas. Do wyznaczenia przynależności do klas zastosowano metody bayesowskie.
EN
ln the 1996-1999 under leadership of prof. Franciszek Grabski has been realised the grant named "Bayesian estimation of reliability parameters and characteristics on the example air and sea systems of transport". Some of the results obtained in the grant were published in the book " Bayesian methods in reliability and diagnosis". The presented here publication contains some of the problems from that book. In the work such concepts like: the state of an object, the state of break down, exposure time, factor of covering are defined. These concepts describe an object under diagnosis. The problems of recognition and their characteristics were considered also. The concept of a class was formulated The logic diagnosis was considered and the concurrence of diagnosis with the concept of the class were shown. The Bayesian methods for determining the collocation with the class were applied.
19
Content available remote The models of non-renewal reliability systems with dependent lengths of components
63%
Grabski F. , Załęska-Fornal A.
|
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tom No. 2
391--398
EN
In that paper there are presented the models of the non-renewal reliability systems with dependent times to failure of components. Their dependence follows from some common sources of shocks existing in the environment. It is assumed that the failure occurs only because of two independent sources common for two neighbour components. The series and parallel system with such dependent components are considered in that paper.
20
Content available Bootstrap methods for the censored data in empirical Bayes estimation of the reliability parameters
63%
Grabski F. , Załęska-Fornal A.
|
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tom Vol. 1
107--112
EN
Bootstrap and resampling methods are the computer methods used in applied statistics. They are types of the Monte Carlo method based on the observed data. Bradley Efron described the bootstrap method in 1979 and he has written a lot about it and its generalizations since then. Here we apply these methods in an empirical Bayes estimation using bootstrap copies of the censored data to obtain an empirical prior distribution.
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