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Content available Maximization problem of three tasks operation process subject to constraint of availability in semi-Markov reliability model
Grabski Franciszek, Rutkowska Maria
Safety and Reliability of Systems and Processes
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2021
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Summer Safety and Reliability Seminar 2021
129--138
EN
Semi-Markov decision processes theory delivers the methods which allow to control the operation processes of the systems. The infinite duration semi-Markov decision processes are presented in the chapter. The gain maximization problem of three tasks operation processes subject to constraint of an availability of the semi-Markov reliability model is discussed. The problem is transformed on some linear programing maximization problem.
2
Content available Nonhomogeneus generalisations of poisson process in the modeling of random processes related to road accident
Grabski Franciszek
Scientific Journal of Polish Naval Academy
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2020
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R. 61 nr 3 (222)
29--42
EN
The stochastic processes theory provides concepts, and theorems, which allow to build the probabilistic models concerning accidents. “Counting process” can be applied for modelling the number of road, sea, and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The nonhomogeneous Poisson process, and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number, and number of people injured and killed in Polish roads. To estimate model parameters were used data coming from the annual reports of the Polish police.
PL
W pracy przedstawiono niektóre 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 probabilistycznego, 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. Estymacja parametrów modelu została wykonana na podstawie danych zamieszczonych w rocznych raportach Policji.
3
Content available Random processes related to accidents on Baltic Sea waters and ports
Grabski Franciszek
Safety and Reliability of Systems and Processes
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2020
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Summer Safety and Reliability Seminar 2020
91--99
EN
A crucial role in construction of the models related to accidents on the 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 sea and seaports are here presented. Moreover some procedures of the models parameters identification are presented in the chapter. Estimation of model some parameters was made based on data from reports of HELCOM and Interreg project Baltic LINes.
4
Content available Random models of accidents at Baltic Port and sea water areas
Grabski Franciszek
Journal of Polish Safety and Reliability Association
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2019
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Vol. 10, No. 1
59--70
EN
The stochastic processes theory provides concepts and theorems that allow to build probabilistic models concerning incidents or (and) accidents in the Baltic Sea waters and ports. A crucial role in construction of the models plays a Poisson process and its extensions; especially a nonhomogeneous Poisson process and nonhomogeneous compound Poisson process. The nonhomogeneous Poisson process allows to build models of accidents number in the sea and seaports. The nonhomogeneous compound Poisson process creates the possibility of constructing models describing the consequences of dangerous events and marine accidents. 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). The expected number of accidents often depends on changing randomly external conditions. Thus it can be assumed that the parameter γ is a random variable. In the paper is assumed that this random variable has a gamma distribution.
5
Content available Nonhomogeneous poisson process and compound poisson process in the modelling of random processes related to road accidents
Grabski Franciszek
Journal of KONES
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2019
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Vol. 26, No. 1
39--46
EN
The stochastic processes theory provides concepts and theorems that allow building probabilistic models concerning accidents. So called counting process can be applied for modelling the number of the road, sea and railway accidents in the given time intervals. A crucial role in construction of the models plays a Poisson process and its generalizations. The new theoretical results regarding compound Poisson process are presented in the paper. A nonhomogeneous Poisson process and the corresponding nonhomogeneous compound Poisson process are applied for modelling the road accidents number and number of injured and killed people in the Polish road. To estimate model parameters were used data coming from the annual reports of the Polish police [9, 10]. Constructed models allowed anticipating number of accidents at any time interval with a length of h and the accident consequences. We obtained the expected value of fatalities or injured and the corresponding standard deviation in the given time interval. The statistical distribution of fatalities number in a single accident and statistical distribution of injured people number and also probability distribution of fatalities or injured number in a single accident are computed. It seems that the presented examples explain basic concepts and results discussed in the paper.
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