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1

Series representation of time-stable stochastic processes

Kopp C., Molchanov I.

Probability and Mathematical Statistics

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2018

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Vol. 38, Fasc. 2

299--315

EN

A stochastically continuous process ξ(t), t ≥ 0, is said to be time-stable if the sum of n i.i.d. copies of ξ equals in distribution the time-scaled stochastic process ξ(nt), t ≥ 0. The paper advances the understanding of time-stable processes by means of their LePage series representations as the sum of i.i.d. processes with the arguments scaled by the sequence of successive points of the unit intensity Poisson process on [0,∞). These series yield numerous examples of stochastic processes that share one-dimensional distributions with a Lévy process.

2

Two lilypond systems of finite line-segments

Daley D. J., Ebert S., Last G.

Probability and Mathematical Statistics

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2016

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Vol. 36, Fasc. 2

221--246

EN

The paper discusses two models for non-overlapping finite line-segments constructed via the lilypond protocol, operating here on a given array of points P = {Pi} in R2 with which are associated directions {θi}. At time zero, for each and every i, a line-segment Li starts growing at unit rate around the point Pi in the direction θi, the point Pi remaining at the centre of Li; each line-segment, under Model 1, ceases growth when one of its ends hits another line, while under Model 2, its growth ceases either when one of its ends hits another line or when it is hit by the growing end of some other line. The paper shows that these procedures are well defined and gives constructive algorithms to compute the half-lengths Ri of all Li. Moreover, it specifies assumptions under which stochastic versions, i.e. models based on point processes, exist. Afterwards, it deals with the question as to whether there is percolation in Model 1. The paper concludes with a section containing several conjectures and final remarks.

3

Stochastic modelling of the repairable system

Andrzejczak K.

Journal of KONBiN

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2015

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No. 3 (35)

5--14

EN

All reliability models consisting of random time factors form stochastic processes. In this paper we recall the definitions of the most common point processes which are used for modelling of repairable systems. Particularly this paper presents stochastic processes as examples of reliability systems for the support of the maintenance related decisions. We consider the simplest one-unit system with a negligible repair or replacement time, i.e., the unit is operating and is repaired or replaced at failure, where the time required for repair and replacement is negligible. When the repair or replacement is completed, the unit becomes as good as new and resumes operation. The stochastic modelling of recoverable systems constitutes an excellent method of supporting maintenance related decision-making processes and enables their more rational use.

PL

Wszelkie modele niezawodności uwzględniające czynniki z losowym czasem przyjmują postać różnych procesów stochastycznych. W tej publikacji przywołujemy definicje podstawowych procesów stosowanych do modelowania systemów naprawialnych. W szczególności przedstawione są procesy stochastyczne wspomagające podejmowanie decyzji dotyczących utrzymania systemu technicznego. Rozważamy najprostszy jednoelementowy system z zaniedbywalnym czasem naprawy lub wymiany, tj. jednostka jest użytkowana a po uszkodzeniu jest naprawiana lub wymieniana w na tyle krótkim czasie, że można go pominąć. Po naprawie lub wymianie jednostka jest tak dobra jak nowa i dalej jest użytkowana. Stochastyczne modelowanie systemów naprawialnych stanowi doskonałą metodę wspomagania procesów decyzyjnych w ich utrzymaniu i pozwala na bardziej racjonalną ich eksploatację.

4

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.

5

Probabilistic methods for estimating the long-term spatial characteristics of monogenetic volcanoes in Japan

Martin A., Takahashi M., Umeda K., Yusa Y.

Acta Geophysica Polonica

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2003

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Vol. 51, nr 3

271-289

EN

The long-term future spatial distribution of monogenetic volcanoes is much more difficult to estimate than for polygenetic volcanoes because the location of the next eruption forming a new edifice is different. Current knowledge of the complex geological factors and natural processes controlling the locations of monogenetic volcanoes is insufficient to estimate future spatial and temporal patterns. This has been achieved instead through the development and application of the probabilistic and statistical approach, especially during the last two decades. In Japan, a lot research has been focused on the construction of volcanic hazard maps in the event of a volcanic eruption for natural disaster management - the consequences of volcanism - but not the probability of new volcanic edifices forming within or nearby volcanic fields for geological disposal. In this paper, the development of the probabilistic approach is described and the results of probabilistic case studies on the Higashi-Izu (59 vents) and the Kannabe -Oginosen (38 vents) Monogenetic Volcano Groups In Japan are reported using: (1) a spatial model; and (2) a spatio-temporal nearest-neighbor model. The accuracy of each model is also examined.

6

Heat transfer crisis on porous surfaces - experimental investigations and the outline of the maximum point process based model

Poniewski M. E., Wiśniewski M., Wójcik T. M.

Zeszyty Naukowe Politechniki Świętokrzyskiej. Mechanika

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1999

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Z. 67

323--338

EN

The paper presented the results of experimental investigations into intralayer boiling crisis and accompanying phenomena. It was observed that the structural parameters of the porous covering affected the course of the process. Hysteresis phenomenon manifested itself when the heat flux initiating intralayer heat crisis was reached. The crisis mechanism hypotheses, the description of which was available in literature, were discussed. The maximum point process was selected as a tool of the mechanism modelling. There was presented the methodology of the calculations of intralayer boiling crisis occurrence probability which depends on physical and chemical properties of the surface and the liquid as well as the porous structure geometrical properties.