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1
Content available Nonhomogeneous stochastic processes connected to Poisson process
Grabski F.
Zeszyty Naukowe Akademii Marynarki Wojennej
|
2018
|
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.
2
Content available Nonhomogeneous compound Poisson process application to modeling of random processes related to accidents in the Baltic Sea waters and ports
Grabski F.
Journal of Polish Safety and Reliability Association
|
2018
|
Vol. 9, No. 3
21--30
EN
A crucial role in construction of the models related to accidents on Baltic Sea water and ports play nonhomogeneous Poisson and nonhomogeneous compound Poisson process. The model of consequences and connected to it model of accidents number on Baltic sea waters and ports are here presented. Moreover some procedures of the models parameters identification are presented in the paper. Estimation of some model parameters was made based on data from reports of HELCOM [10, 11], Interreg project Baltic LINes [9] and EMSA [13].
3
Content available Semi-Markov reliability model of two different units cold standby system
Grabski F.
Zeszyty Naukowe Akademii Marynarki Wojennej
|
2017
|
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.
4
Content available Semi-Markov Reliability Model of System Composed of Main Subsystem, Cold Backup Component and Switch
Grabski F.
Journal of Polish Safety and Reliability Association
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2017
|
Vol. 8, No. 1
47--54
EN
Probabilistic model of a system composed of a main component, an emergency backup component and the automatic switch are discussed in this paper. The reliability model is semi-Markov process describing evolution of the system. Conditional 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. To calculate the reliability function and mean time to failure of the system we apply theorems of the Semi-Markov processes theory concerning the conditional reliability functions.
5
Content available Nonhomogenous Poisson process application to modelling accidents number at Baltic Sea waters and ports
Grabski F.
Journal of Polish Safety and Reliability Association
|
2017
|
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).
6
Content available Semi-Markov model of multi-modal transport operation
Grabski F.
Journal of KONES
|
2017
|
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.
7
Content available remote The renewal process generated by return times of semi-Markov process in reliability models
Grabski F.
Journal of KONBiN
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2017
|
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.
8
Content available Concept of semi-Markov process
Grabski F.
Zeszyty Naukowe Akademii Marynarki Wojennej
|
2016
|
R. 57 nr 3 (206)
25--36
EN
This paper provides the definitions and basic properties related to a discrete state space semi-Markov process. The semi-Markov process is constructed by the so called Markov renewal process that is a special case the two-dimensional Markov sequence. The Markov renewal process is defined by the transition probabilities matrix, called the renewal kernel and an initial distribution or by another characteristics which are equivalent to the renewal kernel. The counting process corresponding to the semi-Markov process allows to determine concept of the process regularity. In the paper are also shown the other methods of determining the semi-Markov process. The presented concepts are illustrated a simple example.
PL
Artykuł przedstawia definicje i podstawowe cechy procesu semi-Markowa dyskretnego stanu przestrzeni. Proces semi-Markova jest zbudowany przez tzw. proces odnawiania Markova, który jest specjalnym przypadkiem dwuwymiarowego ciągu Markova. Proces odnawiania Markova jest zdefiniowany przez macierz prawdopodobieństw przejściowych, zwaną jądrem odnawiania, i początkowy rozkład lub przez inne charakterystyki, które są równe jądru odnawiania. Proces obliczeniowy odpowiadający procesowi semi-Markova pozwala na określenie koncepcji regularności procesu. W artykule przedstawiono również pozostałe metody określania procesu semi-Markova. Przedstawione koncepcje są zaprezentowane na prostym przykładzie.
9
Content available Reliability and maintainability characteristics in semi-Markov models
Grabski F.
Journal of Polish Safety and Reliability Association
|
2016
|
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.
10
Content available Monte Carlo method for reliability of parallel system with dependent failures of component
Grabski F.
Journal of Polish Safety and Reliability Association
|
2016
|
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.
11
Content available Constructing stochastic models for investigation of dangerous events and accidents number in Baltic Sea region ports
Grabski F.
Journal of Polish Safety and Reliability Association
|
2016
|
Vol. 7, No. 1
73--78
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 number of the dangerous events and accidents number in Baltic Sea region ports in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. Three models of the incidents or (and) accidents number in the seaports are here constructed. Moreover some procedures of the model parameters identification and the computer procedures for anticipation of the dangerous events number are presented in the paper.
12
Content available remote Reliability electrical power system of hospital as cold standby system
Grabski F.
Journal of KONBiN
|
2016
|
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.
13
Content available Decision problem for a finite states change of semi-Markov process
Grabski F.
Journal of Polish Safety and Reliability Association
|
2015
|
Vol. 6, No. 1
95--100
EN
In the paper there are presented basic concepts and some results of the theory of semi-Markov decision processes. The algorithm of optimization a SM decision process with a finite number of state changes is discussed here. The algorithm is based on a dynamic programming method. To clarify it the SM decision model for the maintenance operation is shown.
14
Content available Semi-Markov processes: application in system reliability and maintenance
Grabski F.
Journal of Polish Safety and Reliability Association
|
2014
|
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.
15
Content available Non-renewal multistate system with Semi-Markov components
Grabski F.
Journal of Polish Safety and Reliability Association
|
2014
|
Vol. 5, No. 1
63--70
EN
The paper deals with non-renewal multistate monotone systems consisting of multistate components which are modeled by the semi-Markov processes. In the case of a non-renewal system the multistate reliability functions of the system components and the whole system are discussed. All presented concepts and models are illustrated by simple numerical examples.
16
Content available Decision problem for infinite duration semi-Markov process
Grabski F.
Journal of Polish Safety and Reliability Association
|
2014
|
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 Application of perturbed semi-Markov processes in reliability
Grabski F.
Journal of Polish Safety and Reliability Association
|
2013
|
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.
18
Content available Random Failure Rate
Grabski F.
Journal of Polish Safety and Reliability Association
|
2012
|
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.
19
Content available Stochastyczny model bezpieczeństwa obiektu w procesie eksploatacji
Grabski F.
Problemy Eksploatacji
|
2011
|
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.
20
Semi-markowski proces decyzyjny jako model optymalizacji procesu eksploatacji statku
Grabski F.
Logistyka
|
2011
|
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.
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