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1
Content available Influence of transients on end-to-end availability for a meshed network
Tanguy Christian
Safety and Reliability of Systems and Processes
|
2023
|
Summer Safety and Reliability Seminar 2023
207--226
EN
The disruption of services must be kept at a minimum in telecommunications networks so that the consequences are not too severe and their durations are as short as possible. Maintenance policies often rely on the steady-state availabilities of each element of the system, and focus on the system’s weak links. The end-to-end (or two-terminal) availability – a standard performance index – of a meshed network has long been studied, but mainly for small systems, and assuming constant values for the availability of each element. When taken into account, the time-dependent contributions of links and nodes to the system unavailability were computed using exponential failure and repair distributions. In this work we revisit the meshed network first proposed by Walter, Esch, and Limbourg (ESREL 2008), and compute the end-to-end availability between two nodes, where the individual contributions of links and nodes are kept. This allows the ranking of links and nodes, using well-known performance indices (Birnbaum, Risk Reduction Worth, etc.). We can thus determine the elements that should receive due attention in maintenance and resilience studies. However, as the steady-state availability may not always be a lower bound to the transient availability in the case of non-exponential failure and repair distributions, we have studied the influence of such configurations on the time-dependent behaviours of all the aforementioned quantities. We then discuss the influence of uncertainty in the availability values, and compare the results obtained for the all-terminal reliability, another often-used performance criterion of networks.
2
Content available Transient behaviour of instantaneous and average availabilities in non-Markovian configuration
Tanguy Christian
Safety and Reliability of Systems and Processes
|
2022
|
Summer Safety and Reliability Seminar 2022
185--194
EN
In this work, we calculate the exact instantaneous and average availabilities for a system in which the failure distribution is a gamma distribution with a rational shape parameter α, and the repair time distribution is exponential. Various regimes exist, for which the availabilities may or may not attain values below the asymptotic limit. This is an example of configurations where using the steady-state value may lead to an overoptimistic assessment of the availability of an equipment or system during its mission time.
3
Content available remote On Mixtures of Gamma Distributions, Distributions with Hyperbolically Monotone Densities and Generalized Gamma Convolutions (GGC)
Sjödin Tord
Probability and Mathematical Statistics
|
2021
|
Vol. 41, Fasc. 1
1--7
EN
Let Y be a standard Gamma(k) distributed random variable (rv), k > 0, and let X be an independent positive rv. If X has a hyperbolically monotone density of order k (HMk), then Y · X and Y/X are generalized gamma convolutions (GGC). This extends work by Roynette et al. and Behme and Bondesson. The same conclusion holds with Y replaced by a finite sum of independent gamma variables with sum of shape parameters at most k. Both results are applied to subclasses of GGC.
4
Content available remote On background driving distribution functions (BDDF) for some selfdecomposable variables
Jurek Zbigniew J.
Mathematica Applicanda
|
2021
|
Vol. 49, no. 2
85--109
EN
Many classical variables (statistics) are selfdecomposable. They admit the random integral representations via Levy processes. In this note are given formulas for their background driving distribution functions (BDDF). This may be used for a simulation of those variables. Among the examples discussed are: gamma variables, hyperbolic characteristic functions, Student t-distributions, stochastic area under planar Brownian motions, inverse Gaussian variable, logistic distributions, non-central chi-square, Bessel densities and Fisher z-distributions. Found representations might be of use in statistical applications.
PL
Wiele klasycznych modeli probabilistycznych opiera sie o zmienne losowe samorozkładalne. Maja one losowe reprezentacje całkowe oparte o procesy Lévy’ego. W tej notatce podano wzory dla ich kierujących (generujących) dystrybuant. Takich reprezentacji można używać do symulacji tych zmiennych. Wśród omawianych przykładów są: rozkłady t-Studenta, pole stochastyczne pod planarnymi ruchami Browna, odwrotny rozkład Gaussa, rozkłady logistyczne, niecentralny rozkład chi-kwadrat, rozkład Bessela i rozkłady statystyk Z-Fishera. Podane reprezentację mogą być przydatne w statystyce.
5
Content available remote Weyl Multifractional Ornstein–Uhlenbeck Processes Mixed with A Gamma Distribution
Es-Sebaiy Khalifa, Farah Fatima-Ezzahra, Hilbert Astrid
Probability and Mathematical Statistics
|
2020
|
Vol. 40, Fasc. 2
269--295
EN
The aim of this paper is to study the asymptotic behavior of aggregated Weyl multifractional Ornstein–Uhlenbeck processes mixed with Gamma random variables. This allows us to introduce a new class of processes, Gamma-mixed Weyl multifractional Ornstein–Uhlenbeck processes (GWmOU), and study their elementary properties such as Hausdorff dimension, local self-similarity and short-range dependence. We also prove that these processes approach the multifractional Brownian motion.
6
Content available Asymptotic approximations to the non-isothermal distributed activation energy model for bio-mass pyrolysis
Dhaundiyal A., Singh S. B.
Acta Mechanica et Automatica
|
2017
|
Vol. 11, no. 4
293--301
EN
This paper describes the influence of some parameters significant to biomass pyrolysis on the numerical solutions of the non-isothermal nth order distributed activation energy model (DAEM) using the Gamma distribution and discusses the special case for the positive integer value of the scale parameter (𝜆), i.e. the Erlang distribution. Investigated parameters are the integral upper limit, the frequency factor, the heating rate, the reaction order, and the shape and rate parameters of the Gamma distribution. Influence of these parameters has been considered for the determination of the kinetic parameters of the non-isothermal nth order Gamma distribution from the experimentally derived thermoanalytical data of biomass pyrolysis. Mathematically, the effect of parameters on numerical solution is also used for predicting the behaviour of the unpyrolysized fraction of biomass with respect to temperature. Analysis of the mathematical model is based upon asymptotic expansions, which leads to the systematic methods for efficient way to determine the accurate approximations. The proposed method, therefore, provides a rapid and highly effective way for estimating the kinetic parameters and the distribution of activation energies.
7
Content available Profit optimalization in operation systems
Knopik L., Migawa K., Wdzięczny A.
Polish Maritime Research
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2016
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nr 1
93--98
EN
The aim of this study is to analyse the effect of reducing the number of secondary damage to profit from the work of a technical object. Considering the criterion function that describes the average profit from the work of a technical object in the operation system. The study analyses a model of profit optimization, in which the design criterion function is based on the properties of Poisson branch process. Criterion function that describes the average gain is considered in the work at sufficiently general factors. Profit lifting model from the work of a technical facility is numerically exemplified. For the analysed electrical subsystem, intervals of time between the initial damage have exponential distribution, and between secondary damage - gamma distribution. In the presented example, the ability of profit optimization in operation systems of technical objects is demonstrated.
8
Content available A control chart using belief information for a gamma distribution
Aslam M., Khan N., Jun C. -H.
Operations Research and Decisions
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2016
|
Vol. 26, No. 4
5--19
EN
The design of a control chart has been presented using a belief estimator by assuming that the quantitative characteristic of interest follows the gamma distribution. The authors present the structure of the proposed chart and derive the average run lengths for in-control and a shifted process. The aver-age run lengths for various specified parameters have been reported. The efficiency of the proposed chart has been compared to existing control charts. The application of the proposed chart is illustrated with the help of simulated data.
9
Model procesu uszkodzeń
Knopik L., Migawa K.
Logistyka
|
2015
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nr 4
1906--1914, CD2
PL
Rozkłady prawdopodobieństwa czasów do uszkodzenia są intensywnie wykorzystywane w teorii i praktyce niezawodności. W praktyce spotyka się wiele sytuacji testowania zgodności rozkładów prawdopodobieństwa czasów do uszkodzenia ze znanymi rozkładami. Jednak w praktyce eksploatacyjnej bardzo często zdarzają się sytuacje, gdy obiekt techniczny uszkadza się nagle i rejestrowane czasy do uszkodzenia są stosunkowo małe. Naprawy niektórych uszkodzeń są nieskuteczne i generują następne uszkodzenia. Zbiór czasów do uszkodzenia wykazuje pewną niejednorodność statystyczną. Jest to motywacją do zastosowania jako modelu procesu uszkodzeń stacjonarnego gałązkowego procesu Poissona. W pracy dowodzi się, że proces ten może być adekwatnym modelem procesu uszkodzeń dla podsystemu elektrycznego autobusów. Prezentowany model został zilustrowany przykładami numerycznymi. Drugi z przykładów prezentuje dane z eksploatacji autobusów miejskich i zawiera ocenę parametrów modelu dla podsystemu elektrycznego.
EN
The probability distributions of time to failure are used extensively in the theory and practice of reliability. In practice, there are many situations of testing compatibility of time to failure probability distributions with known distributions. However, in practice, there are often situations when the technical object suddenly becomes failures the recorded times to failures relatively small. Repairs of some failures are ineffective and generate more failures. The set of times to failures has a certain statistical heterogeneity. This is the motivation as a model of failure process using branching Poisson process. The paper argues that this process may be an adequate model of the process of failures to the electrical subsystem of buses. This model is illustrated by numerical examples. The second example shows the data from the maintenance of buses and includes an assessment of the model parameters for the electrical subsystem.
10
Content available Evaluation of the Precision of Standardized Precipitation Index (SPI) Based On Years 1954-1995 in Łódź
Gąsiorek E., Musiał E.
Journal of Ecological Engineering
|
2015
|
Vol. 16, nr 4
49--53
EN
This paper evaluates the accuracy of estimates of the Standardized Precipitation Index (SPI) using gamma, normal and log-normal distributions. In order to classify the above methods, the authors performed an analysis of the quality of theoretical distributions to empirical distribution, obtained on the basis of monthly precipitation sums during the vegetation season in a multi-year period 1954–1995 in Łódź.
11
Content available Wyznaczanie wskaźnika standaryzowanego opadu (SPI) z zastosowaniem rozkładu gamma
Gąsiorek E., Musiał E.
Przegląd Naukowy Inżynieria i Kształtowanie Środowiska
|
2014
|
Vol. 23, No. 3
237--249
PL
W prezentowanej pracy autorzy skorzystali z danych pomiarowych pochodzących ze stacji Instytutu Technologiczno-Przyrodniczego w Bydgoszczy w wieloleciu 1946–2003 i przeprowadzili analizę zgodności otrzymanych wartości wskaźników standaryzowanego opadu (SPI) otrzymanych dwiema metodami. Pierwsza z nich polega na wyznaczeniu SPI bezpośrednio z dopasowanego rozkładu gamma, któremu podlegają miesięczne sumy opadów w wieloleciu 1946–2003 w Bydgoszczy, a druga na zastosowaniu transformacji prowadzących do rozkładu normalnego. Celem tej pracy była weryfikacja zaproponowanych wcześniej metod na nowym materiale empirycznym. Z dwóch stosowanych metod, metoda bezpośredniego wyznaczania wskaźnika SPI z rozkładu gamma jest mniej pracochłonna i daje lepszą ocenę warunków opadowych niż metoda polegająca na znajdowaniu transformacji przekształcających rozkład gamma w rozkład normalny.
EN
In the current paper the authors have used the data from the Institute of Technology and Natural Sciences observatory in Bydgoszcz in 1946–2003 period. They have analyzed the consistency of standardized precipitation index (SPI) values, calculated with the use of two different methods. The first one extracts SPI directly from gamma distribution, with the assumption that monthly precipitation sums in 1946–2003 in Bydgoszcz are gamma distributed. The second method is based on the transformations of data leading to normal distribution. The aim of this paper was to verify the previously proposed methods on new empirical data. Out of those two methods, the direct calculation of SPI from gamma distribution is less time consuming and gives better evaluation of precipitation conditions than the method based on transformation from gamma to normal distribution.
12
Content available Statistical downscaling of probability density function of daily precipitation on the Polish coast
Wójcik R., Pilarski M., Miętus M.
Meteorology Hydrology and Water Management. Research and Operational Applications
|
2014
|
Vol. 2, Iss. 1
27--36
EN
The aim of this study was to recognize the possibility of downscaling probability density function (PDF) of daily precipitation by means of canonical correlation analysis (CCA). Sea level pressure (SLP) over Europe and the North Atlantic was used as a predictor. A skilful statistical model could be used to generate projections of future changes of precipitation PDF driven by GCM (General Circulation Model) simulations. Daily precipitation totals from 8 stations located on the Polish coast of the Baltic Sea covering the period 1961-2010 were used to estimate the gamma distribution parameters, and only wet days (i.e. ≥0.1mm) were taken in the analysis. The results of the Kolmogorov-Smirnov test and comparison of empirical and theoretical (gamma-distributed) quantiles proved that gamma distribution gives a reliable description of daily precipitation totals. The validation of CCA models applied to gamma parameters revealed that the reliable reconstruction of precipitation PDF is possible only for average long-term conditions. In the case of individual months/seasons the agreement between empirical and reconstructed quantiles is poor. This study shows the potential of modelling of precipitation PDF, however efforts should be made to improve model performance by establishing more reliable links between regional forcing and the variability of the gamma parameters.
13
Content available Wskaźnik standaryzowanego opadu (SPI) wyznaczony z zastosowaniem rozkładu log-normalnego
Gąsiorek E., Musiał E., Rojek M.
Inżynieria Ekologiczna
|
2014
|
Nr 39
62--73
PL
Praca jest kontynuacją badań przeprowadzonych na podstawie danych z Obserwatorium Agro- i Hydrometeorologii UP Wrocław-Swojec z okresu 1964–2009 i opublikowanych w czasopiśmie „Infrastruktura i Ekologia Terenów Wiejskich” nr 3/III/2012, str. 197–208. Tematyką opublikowanej pracy są dwie metody wyznaczania wskaźnika standaryzowanego opadu. Pierwsza z nich polega na wyznaczeniu SPI bezpośrednio z dopasowanego rozkładu gamma, któremu podlegają miesięczne sumy opadów w wieloleciu 1964–2009 we Wrocławiu, druga na zastosowaniu transformacji prowadzących do rozkładu normalnego. W prezentowanej pracy autorzy wyznaczają wskaźnik standaryzowanego opadu (SPI) stosując rozkład logarytmiczno-normalny i dokonują oceny tak wyznaczonego wskaźnika na tle wskaźników wyznaczonych za pomocą rozkładu gamma i rozkładu normalnego. Celem pracy była analiza porównawcza otrzymanych wartości współczynników SPI otrzymanych trzema różnymi metodami
EN
The problem analyzed in this paper is the continuation of research conducted on data from Wrocław-Swojec agro- and hydrometeorology observatory in 1964–2009 period and published in “Infrastruktura i Ekologia Terenów Wiejskich” nr 3/III/2012, pp. 197–208. The paper concerns two methods of calculation of standardized precipitation index (SPI). The first one extracts SPI directly from gamma distribution, since monthly precipitation sums in the 1964–2009 period in Wrocław are gamma distributed. The second method is based on the transformations of data leading to normal distribution. The authors calculate SPI with the use of log-normal distribution and confront it with values obtained by gamma and normal distributions. The aim of this paper is to comparatively assess the SPI values obtained with those three different methods.
14
Content available Ocena dokładności oszacowania wskaźnika standaryzowanego opadu (SPI) wyznaczonego za pomocą różnych rozkładów prawdopodobieństwa
Gąsiorek E., Musiał E., Rojek M.
Inżynieria Ekologiczna
|
2014
|
Nr 40
92--99
PL
Stosowanie różnych metod obliczania wskaźnika standaryzowanego opadu (SPI) związane jest z otrzymywaniem różnych jego przybliżeń. Metody oparte na rozkładzie normalnym i jego transformacjach oraz na rozkładzie gamma dają podobne wyniki, stąd można je stosować zamiennie. Metoda oparta na rozkładzie logarytmiczno-normalnym daje wyraźnie różne wartości tego wskaźnika szczególnie dla wartości ekstremalnych SPI. Pojawia się wobec tego problem, która z metod daje rozkład lepiej dopasowany do danych empirycznych. Celem tej pracy jest uporządkowanie powyższych metod ze względu na stopień dopasowania do danych empirycznych z Obserwatorium Agro- i Hydrometeorologii we Wrocławiu Swojcu z wielolecia 1964–2009.
EN
The use of different calculating methods to compute the standardized precipitation index (SPI) results in various approximations. Methods based on normal distribution and its transformations, as well as on gamma distribution, give similar results and may be used equally, whereas the lognormal distribution fitting method is significantly discrepant, especially for extreme values of SPI. Therefore, it is problematic which method gives the distribution optimally fitted to empirical data. The aim of this study is to categorize the above mentioned methods according to the degree of approximation to empirical data from the Observatory of Agro- and Hydrometeorology in Wrocław-Swojec from 1964–2009 years.
15
Content available remote An experimental comparison of popular estimation methods for the Weibull, gamma and Gompertz distributions
Żerda I.
Schedae Informaticae
|
2011
|
Vol. 20
67-82
EN
The aim of this study was to describe some parametric estimation methods for the Weibull, gamma and Gompertz distributions and to identify among them estimators the most efficient in practical applications. Techniques which are considered as traditional methods, like the maximum likelihood (MLE) and the method of moments (MM) estimation but also some newer and less commonly used techniques like the Lmoment estimator (LME), least-square estimator (LSE), generalized spacing estimator (GSE) and percentile estimator (PE) were presented. The application of each method was demonstrated in a simulation study using data sets generated for different distribution parameters and sample sizes. Discussed estimators were compared in terms of their efficiency and bias measured by mean-square errors (MSE) based on the simulations results.
16
Content available remote Several forms of stochastic integral representations of gamma random variables and related topics
Aoyama T., Maejima M., Ueda Y.
Probability and Mathematical Statistics
|
2011
|
Vol. 31, Fasc. 1
99--118
EN
Gamma distributions can be characterized as the laws of stochastic integrals with respect to many different Lévy processes with different nonrandom integrands. A Lévy process corresponds to an infinitely divisible distribution. Therefore, many infinitely divisible distributions can yield a gamma distribution through stochastic integral mappings with different integrands. In this paper, we pick up several integrands which have appeared in characterizing well-studied classes of infinitely divisible distributions, and find inverse images of a gamma distribution through each stochastic integral mapping. As a by-product of our approach to stochastic integral representations of gamma random variables, we find a remarkable new general characterization of classes of infinitely divisible distributions, which were already considered by James et al. (2008) and Aoyama et al. (2010) in some special cases.
17
Content available Zmienność parametrów rozkładu gamma wielkości kropel deszczu naturalnego dla różnych intensywności opadu
Kamińska J., Machowczyk A., Szewrański Sz.
Woda-Środowisko-Obszary Wiejskie
|
2010
|
T. 10, z. 4
95-102
PL
W pracy przedstawiono wyniki analiz zależności między parametrami rozkładu kropel deszczu z zastosowaniem funkcji gamma dla 16-miesięcznego okresu badawczego oraz charakterystykę rozkładów gamma w zależności od intensywności opadu. Charakterystykę opadu deszczu naturalnego wyznaczono w oparciu o bezpośrednie pomiary średnicy i prędkości opadania pojedynczych kropel deszczu wykonane z wykorzystaniem disdrometru laserowego firmy Thies-Clima. Rejestracja danych pluwiometrycznych odbywała się w 1-minutowych interwałach czasowych na terenie Obserwatorium Agro- i Hydrometeorologii we Wrocławiu -Swojcu.
EN
Natural rainfall characteristics had been described based on direct measurements of raindrops diameters and its terminal velocity with the use of laser disdrometer made by Thies-Clima. Pluviometric data were recorded every 1 minute. Monitoring was carried out in Agro- and Hydrometeorological Observatory of the Wrocław University of Environmental and Life Sciences. The variation of gamma distribution parameters for different DSD of natural rainfalls from 16 months study period is presented in this article. The gamma distribution analyses were also made for different rainfall intensities.
18
Content available remote Repairable systems availability optimization under imperfect maintenance
Samet S., Chelbi A., Hmida F.B.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2009
|
Vol. 57, nr 3
249-256
EN
This paper deals with the modeling of a preventive maintenance strategy applied to a single-unit system subject to random failures. According to this policy, the system is subjected to imperfect periodic preventive maintenance restoring it to 'as good as new1 with probability p and leaving it at state 'as bad as old' with probability q. Imperfect repairs are performed following failures occurring between consecutive preventive maintenance actions, i.e the times between failures follow a decreasing quasi-renewal process with parameter a. Considering the average durations of the preventive and corrective maintenance actions as well as their respective efficiency extents, a mathematical model is developed in order to study the evolution of the system stationary availability and determine the optimal PM period which maximizes it. The modeling of the imperfection of the corrective maintenance actions requires the knowledge of the quasi-renewal function. A new expression approximating this function is proposed for systems whose times to first failure follow a Gamma distribution. Numerical results arc obtained and discussed.
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