Aim. Common cause failures (CCFs) are dependent failures of groups of certain elements that occur simultaneously or within a short period of time (i.e. almost simultaneously) due to a single common cause (e.g. a sudden change of climatic operating conditions, flooding of premises, etc.). A dependent failure is a multiple failure of several elements of a system, whose probability cannot be expressed as a simple product of the probabilities of unconditional failures of individual elements. ССА probabilities calculation uses a number of common models, i.e. the Greek letter model, alpha, beta factor and their variants. The beta-factor model is the most simple in terms of simulation of dependent failures and further dependability calculations. Other models, when used in simulation, involve combinatorial enumeration of dependent events in a group of n events that becomes labour-intensive if the number n is high. For the selected structure diagrams of dependability, the paper analyzes the calculation method of system failure probability with CCF taken into account for the beta-factor model. The Aim of the paper is to thoroughly analyze the beta-factor method for three structure diagrams of dependability, research the effects of the model parameters on the final result, find the limitations of beta-factor model applicability. Methods. The calculations were performed using numerical methods of solution of equations, analytical methods of function studies. Conclusions. The paper features an in-depth study of the method of undependability calculation for three structure diagrams that accounts for CCF and uses the beta-factor model. In the first example, for the selected structure diagram out of n parallel elements with identical dependability, it is analytically shown that accounting for CCF does not necessarily cause increased undependability. In the second example of primary junction of n elements with identical dependability, it is shown that accounting for CCF subject to parameter values causes both increased and decreased undependability. A number of beta factor model parameter values was identified that cause unacceptable values of system failure probability. These sets of values correspond to relatively high model parameter values and are hardly practically attainable as part of engineering of real systems with highly dependable components. In the third example, the conventional bridge diagram with two groups of CCFs is considered. The complex ambivalent effect of beta factor model parameters on the probability of failure is shown. As in the second example, limitations of the applicability of the beta-factor model are identified.

Аннотация. Цель. Анализ дерева отказов (АДО) -это один из основных методов анализа надежности сложных технических систем. Для его проведения часто применяются коммерческие программные средства, такие как Saphire, Risk Spectrum, Арбитр и т.д. Каждый из них обладает как определенными преимуществами, так и отдельными недостатками. Необходимо отметить, что основная цель вышеуказанных программных средств состоит в проведении качественного анализа дерева отказов. При этом в качестве дополнительных возможностей программного комплекса предлагается ряд статистических методов, позволяющих, в частности, проводить анализ неопределенностей, получать интервальные оценки показателей, выполнять прочие статистические исследования. Набор таких процедур невелик и жестко регламентирован некоторым множеством предлагаемых распределений и функций. В данной работе рассмотрим возможность решения задачи анализа дерева отказов с помощью средств языка программирования R. Язык программирования R, в первую очередь, создавался и продолжает совершенствоваться как средство статистической обработки данных. АДО в этой среде всего лишь один из 10 с лишним тысяч пакетов. Т.е., если проводить сравнение с коммерческими пакетами, направленность которых состоит в АДО, R ставит перед собой гораздо более масштабные цели и обладает возможностями проведения существенно более качественного анализа. При этом несомненное преимущество R -это свободно распространяемая среда с открытым программным кодом. Цель данной статьи состоит в представлении небольшого числа основных процедур пакета FaultTree языка R, позволяющих проводить АДО: построение дерева отказов и его графический вывод, расчет вероятностей по узлам и нахождение минимальных сечений. Методы. Для выполнения расчетов и демонстрации возможностей АДО применялись скрипты пакета FaultTree языка программирования R. Выводы. В статье подробно разобраны три примера. Вначале рассчитывается дерево по известным вероятностям, затем определяется функция распределения наработки до отказа технической системы. В последнем примере выполняется АДО для систем с элементами, которые описываются различными моделями функционирования и обслуживания. В заключительной части статьи предполагается описание возможностей АДО в среде R позволяющих учитывать, к примеру, отказы по общей причине. Ключевые слова: дерево отказов, анализ дерева отказов, коэффициент неготовности, явные отказы, скрытые отказы, средняя наработка на отказ.

Aim. This paper is the continuation of [1] that proposes using the R programming language for fault tree analysis (FTA). In [1], three examples are examined: fault tree (FT) calculation per known probabilities, dynamic FT calculation per known distributions of times to failure for a system’selements. In the latter example, FTA is performed for systems with elements that are described by different functional and service models. Fault tree analysis (FTA) is one of the primary methods of dependability analysis of complex technical systems. This process often utilizes commercial software tools like Saphire, Risk Spectrum, PTC Windchill Quality, Arbitr, etc. Practically each software tool allows calculating the dependability of complex systems subject to possible common cause failures (CCF). CCF are the associated failures of a group of several elements that occur simultaneously or within a short time interval (i.e. almost simultaneously) due to one common cause (e.g. a sudden change in the climatic service conditions, flooding of the premises, etc.). An associated failure is a multiple failure of several system elements, of which the probability cannot be expressed simply as the product of the probabilities of unconditional failures of individual elements. There are several generally accepted models used in CCF probability calculation: the Greek letters model, the alpha, beta factor models, as well as their variations. The beta factor model is the most simple in terms of associated failures simulation and further dependability calculation. The other models involve combinatorial search associated events in a group of n events, that becomes labor-consuming if the number n is large. Therefore, in the above software tools there are some restrictions on the n, beyond which the probability of CCF is calculated approximately. In the current R FaultTree package version there are no above CCF models, therefore all associated failures have to be simulated manually, which is not complicated if the number of associated events is small, as well as useful in terms of understanding the various CCF models. In this paper, for the selected diagram a detailed analysis of the procedure of associated failures simulation is performed for alpha and beta factor models. The Purposeof this paper consists in the detailed analysis of the alpha and beta factor methods for a certain diagram, in the demonstration of fault tree creation procedure taking account of ССF using R’s FaultTree package. Methods. R’s FaultTree scripts were used for the calculations and FTA capabilities demonstration.Conclusions. Two examples are examined in detail. In the first example, for the selected block diagram that contains two groups of elements subject to associated failures, the alpha factor model is applied. In the second example, the beta factor model is applied. The deficiencies of the current version of FaultTree package are identified. Among the main drawbacks we should indicate the absence of some basic logical gates.

Aim. To modify the classical method [1, 4] that causes incorrect estimation of the required size of SPTA in cases when the replacement rate of failed parts is comparable to the SPTA replenishment rate. The modification is based on the model of SPTA target level replenishment. The model considers two situations: with and without the capability to correct requests in case of required increase of the size of replenishment. The paper also aims to compare the conventional and adjusted solution and to develop recommendations for the practical application of the method of SPTA target level replenishment. Methods. Markovian models [2, 3, 5] are used for describing the system. The flows of events are simple. The final probabilities were obtained using the Kolmogorov equation. The Kolmogorov system of equations has a stationary solution. Classical methods of the probability theory and mathematical theory of dependability [6] were used.  Conclusions. The paper improves upon the known method of estimating the required size of the SPTA with a safety stock. The paper theoretically substantiates the dependence of the rate of backward transitions on the graph state index. It is shown that in situations when the application is not adjusted, the rates of backward transitions from states in which the SPTA safety stock has been reached and exceeded should gradually increase as the stock continues to decrease. The multiplier will have a power-law dependence on the transition rate index. It was theoretically and experimentally proven that the classical method causes SPTA overestimation. Constraint (3) was theoretically derived, under which the problem is solved sufficiently simply using the classical methods. It was shown that if constraint (3) is not observed, mathematically, the value of the backward transition rate becomes uncertain. In this case, correct problem definition results in graphs with a linearly increasing number of states, thus, by default, the problem falls into the category of labour-intensive. If the limits are not observed, a simplifying assumption is made, under which a stationary solution of the problem has been obtained. It is shown that, under that assumption, the solution of the problem is conservative. It was shown that, if the application is adjusted, the rate of backward transition from the same states should gradually decrease as the stock diminishes. The multiplier will have a hyperbolic dependence on the transition rate index. This dependence results in a conservative solution of the problem of replenishment of SPTA with application adjustment. The paper defines the ratio that regulates the degree of conservatism. It is theoretically and experimentally proven that in such case the classical method causes SPTA underestimation. A stationary solution of the problem of SPTA replenishment with application adjustment has been obtained. In both cases of application adjustment reporting, a criterion has been formulated for SPTA replenishment to a specified level. A comparative analysis of the methods was carried out.