Wyniki wyszukiwania - BazTech

1

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.

2

An apartment problem

Szajowski K.

Mathematica Applicanda

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2014

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Vol. 42, No. 2

193--206

EN

In the 60 - ies of the last century, several optimization problems referring to the sequential methods were investigated. These tasks may include the Robbins’ problem of optimal stopping, the secretary problem (see the discussion paper by Ferguson [18]), the parking problem or the job search problem. Subtle details of the wording in these issues cause that each of these terms include family of problems that differ significantly in detail. These issues focused attention of a large group of mathematicians. One of the related topic has been the subject of Professor Jerzy Zabczyk attention. Based on the discussions with Professor Richard Cowan1 the model of choosing the best facility available from a random number of offers was established. In contemporary classification of the best choice problems it is the noinformation, continuous time, secretary problem with the Poisson stream of options and the finite horizon.

PL

W latach 60 -tych poprzedniego wieku analizowano wielu matematyków skupiało swoja uwagę na zadaniach optymalizacyjnych nawiązujących do sekwencyjnego przeszukiwania czy obserwacji. Do tych zadań można zaliczyć problem optymalnego zatrzymania Robbinsa, problem sekretarki, (dość obszerną analizę tego zagadnienia przeprowadził Ferguson [18]), zadanie optymalnego parkowania czy też problem poszukiwania pracy. Subtelne szczegóły tych zagadnień powodują, iż każde zagadnienie z wymienionych ma liczne wersje różniące się szczegółami, które powodują, iż mamy do czynienia całą rodziną modeli. Jedno z zagadnień zainteresowało profesora Jerzy Zabczyk. W wyniku dyskusji z profesorem Richardem Cowanem (w Warszawie ) stworzyli model poszukiwania najlepszego obiektu, gdy dostępnych obiektów jest losowa liczba. Wg współczesnej klasyfikacji problemów wyboru najlepszego obiektu jest to przypadek poszukiwania najlepszego obiektu przy braku informacji, z czasem ciągłym, gdy strumień zgłoszeń jest poissonowski a horyzont jest skończony, ustalony.

3

Distributional properties of the negative binomial Lévy process

Kozubowski T. J., Podgórski K.

Probability and Mathematical Statistics

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2009

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Vol. 29, Fasc. 1

43--71

EN

The geometric distribution leads to a Lévy process parameterized by the probability of success. The resulting negative binomial process (NBP) is a purely jump and non-decreasing process with general negative binomial marginal distributions. We review various stochastic mechanisms leading to this process, and study its distributional structure. These results enable us to establish strong convergence of the NBP in the supremum norm to the gamma process, and lead to a straightforward algorithm for simulating sample paths. We also include a brief discussion of estimation of the NPB parameters, and present an example from hydrology illustrating possible applications of this model.