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1

Estimation of Weibull distribution parameters based on sequences of minimal repairs

Malinowski Jacek

Safety and Reliability of Systems and Processes

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2022

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Summer Safety and Reliability Seminar 2022

163--172

EN

A new method of estimating the scale and shape parameters of the Weibull distribution is presented. According to this method, a Weibull distributed time-to-failure (TTF) of a test item is measured m times. It undergoes a minimal repair after each of the first m-1 failures, and is put out of use after the m-th failure. This procedure is repeated n times. Based on m TTFs of one test item, which are neither independent nor identically distributed (IID), the maximum likelihood estimators (MLE) of the scale and shape parameters, called n m-sample estimators, are obtained. The accuracy of the m-sample estimators is low, however, it can be improved by using the mean values of their n IID realizations as more precise estimators. The latter are called n·m-sample estimators, have the same biases as the respective m-sample ones, but their variances are n times smaller. Interestingly enough, the n·m-sample estimators of the scale and shape parameters, as well as their biases, are given by relatively simple explicit formulas. This is somewhat unexpected in view of the fact that the standard MLE of the shape parameter, based on IID TTFs of non-repairable test items, is obtained from an equation that cannot be solved analytically.

2

Comparative analysis of Riemann-Liouville, Caputo-Fabrizio, and Atangana-Baleanu integrals

Shaikh Asif Ali, Qureshi Sania

Journal of Applied Mathematics and Computational Mechanics

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2022

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Vol. 21, nr 1

91--101

EN

This study analyzes the most commonly used operators of the Riemann-Liouville, the Caputo-Fabrizio, and the Atangana-Baleanu integral operators. Firstly, a numerical scheme for the Riemann-Liouville fractional integral has been discussed. Then, two numerical techniques have been suggested for the remaining two operators. The experimental order of convergence for the schemes is further supported by the computations of absolute relative error at the final nodal point over the integration interval [0, T ]. Comparative analysis of the integrals reveals that the Riemann-Liouville fractional integral yields the most minor errors and the most significant experimental order of convergence in the majority of functions, particularly when the fractional-order parameter α → 0. It is worth noting that the Atangana-Baleanu has proved to be an essential operator for solving many dynamical systems that a single RL operator cannot handle. All of the three integral operators coincide with each other for α = 1. Mathematica 11.3 for an Intel(R) Core(TM) i3-4500U procesor running on 1.70 GHz is used to carry out all the necessary computations.

3

Characterizations of compact operators on ℓp-type fractional sets of sequences

Özger Faruk

Demonstratio Mathematica

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2019

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Vol. 52, nr 1

105--115

EN

Among the sets of sequences studied, difference sets of sequences are probably the most common type of sets. This paper considers some ℓp-type fractional difference sets via the gamma function. Although, we characterize compactness conditions on those sets using the main key of Hausdorff measure of noncompactness, we can only obtain sufficient conditions when the final space is ℓ∞. However, we use some recent results to exactly characterize the classes of compact matrix operators when the final space is the set of bounded sequences.

4

An inequality involving the gamma and digamma functions

Qi F., Guo B.-N.

Journal of Applied Analysis

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2016

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Vol. 22, nr 1

49--54

EN

In the paper, the authors establish an inequality involving the gamma and digamma functions and apply it to prove the negativity and monotonicity of a function involving the gamma and digamma functions.

5

Some intriguing limits - continuation

Lorenc P., Wituła R.

Zeszyty Naukowe. Matematyka Stosowana / Politechnika Śląska

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2013

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z. 3

17--24

EN

Wituła and Słota in [College Math. J. 42 (2011), 328] proposed a way of proving the relation (1) given below which appeared to be a genuine result. Authors of the present paper, inspired by the form of this limit, try to find some generalizations of this one, also in the context of some special functions (e.g. the gamma function, the generalized Laguerre polynomials).

PL

Wituła i Słota w notce [College Math. J. 42 (2011), 328] zaproponowali udowodnienie relacji (1), podanej poniżej, która wydaje się bardzo ciekawą zależnością. Autorzy niniejszego artykułu, zainspirowani postacią tej granicy, próbują znaleźć różne jej uogólnienia także w kontekście pewnych funkcji specjalnych (np. funkcji gamma, uogólnionych wielomianów Laguerre’a).

6

Inequalities for Γp function and gamma function

Yin L., Song Z. M.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

485--490

EN

In this paper, we present a double inequality for the gamma function by estimating bounds of Γp function. Later, we also give a new inequality of gamma function.

7

Applications of convolution properties

Trojnar-Spelina L.

Demonstratio Mathematica

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2013

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Vol. 46, nr 3

513--523

EN

K. I. Noor (2007 Appl. Math. Comput. 188, 814–823) has defined the classes Qk(a, b, λ, γ) and Tk(a, b, λ, γ) of analytic functions by means of linear operator connected with incomplete beta function. In this paper, we have extended some of the results and have given other properties concerning these classes.

8

O modelu Strejca z ułamkowym rzędem inercji w dziedzinie czasu

Wesołowski J.

Zeszyty Naukowe Politechniki Rzeszowskiej. Elektrotechnika

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2004

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z. 27 [220]

211--217

PL

Niniejsza publikacja przedstawia koncepcję tworzenia modelu Strejca z ułamkowym rzędem inercji wyrażonym za pomocą funkcji sklejanych. Podane zostały zarówno inne postacie matematycznej formy opisu modelu jak również przykładowy algorytm jaki można zaimplementować w programie MatLab.

EN

The article presents a concept of constructing Strejc's model with fractional inertia order using the combined functions. The article includes different forms of mathematical description of the model and an example algorithm suitable for implementation under the computer program MatLab.

9

Charakterystyki systemu M/Wk/1

Filipowicz B., Rutkowska A.

Elektrotechnika i Elektronika

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2001

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T. 20, z. 1

14--21

PL

Artykuł jest próbą poznania własności systemu kolejkowego z poissonowskim strumieniem zgłoszeń, jednym serwerem i naturalnym regulaminem kolejki, o czasie obsługi będącym zmienną losową o rozkładzie Weibulla. Pierwszym krokiem było poznanie własności gęstości prawdopodobieństwa, jej transformaty Laplace'a oraz momentów tego rozkładu. Kolejny etap obejmował badanie zachowania charakterystyk systemu przy użyciu wzorów Pollaczka -Chinczyna oraz własności funkcji Γ. Nieco uwagi poświęcono również analizie procesu wyjściowego. Ostatnim etapem było zbudowanie modelu symulacyjnego dla omawianego systemu oraz analiza statystyczna wyników. Omówiony model może służyć jako metoda badania systemu z „męczącym się" serwerem występującym w różnych dziedzinach życia.

EN

In the paper there is dicussed the system with a single - server queue, a Poisson input and Weibull service time distribution. In the first part theproperties of the Weibull density together with his Laplace transform and his moments are presented. Then there are analysed the characteristics of the system; the Pollaczek—Chinczyn formulas and the properties of the Γ function are helpful in studying their behaviour. There are also discussed some analytic results of the output process. At the end of the paper a simulation model is presented and then the statistical data analysis of the characteristics and of the output process. This model can be usefiil in studying the behaviour of the system with the server, that "fatigues ", applied in many areas of research.