Event tree analysis basic calculations

Tliis cliapter is concerned willi special probability distributions and tecliniques used in calculations of reliability and risk. Tlieorems and basic concepts of probability presented in Cliapter 19 are applied to llie determination of llie reliability of complex systems in terms of tlie reliabilities of their components. Tlie relationship between reliability and failure rate is explored in detail. Special probability distributions for failure time are discussed. Tlie chapter concludes with a consideration of fault tree analysis and event tree analysis, two special teclmiques lliat figure prominently in hazard analysis and llie evaluation of risk. 

Risk analysis is required to evaluate the accident frequency and consequences. In railway industry. Safety Risk Model (SRM) is used to estimate system risk, SRM consists of Fault Tree Analysis (FTA) and Event Tree Analysis (ETA). Fault tree estimates accident frequency considering system failure logic (Muttram 2002). It calculates top event frequency or probability using minimal cut sets. Basic events in fault tree describe the component failures they can model revealed repairable failure, revealed unrepairable failure and unrevealed repairable failure with periodic inspection (Andrews Moss 2002). The above failure models for basic event are not enough to consider the effects of maintenances on risk as these models cannot describe multi-level repairsor inspections in details. 

Fault Tree. When direct data allowing to calculate the probability of a failure mode are not available or this failure form is complex, it is proposed the elaboration of a fault tree. It is a method of multidisciplinar analysis that begins with the selection of a failure mode or event that is tried to avoid. The event is developed into its immediate causes, and the sequence of events continues until basic causes are identified. The fault tree is constructed showing the logical event relationships that are necessary to result in the top event. The fault tree reaches terminal events whose probability must be calculated or estimated. These events can be basic events, which do not require to be explained by means of other previous events, or events which are not developed because it is not considered necessary or for lack of information. 

Using fiizzy interval analysis, a possibility distribution for the top event probability (chance), 7r(q), is obtained by calculating lower and upper a-cut values of the top event probability (chance) using the corresponding lower and upper a-cut values for the basic event probabilities (chances). The reader is referred to Singer (1990) for further details on the fiizzy set approach to fault tree analysis. 

A THERP tree is a technique used in human reliability assessment to calculate the probability of a human error during the execution of a task. (THERP stands for Technique for Human Error Rate Prediction.) A THERP tree is basically an event tree, where the root is the initiating event and the leaves are the possible outcomes. THERP is described in a publication from 1983 (Swain, A.D. and Guttmann, H.E., Handbook of Human Reliability Analysis with Emphasis on Nuclear Power Plant Applications, NUREG/CR-1278, USNRC), and is still widely used despite its unrealistic assumptions about human performance. One important... 

Singer applied fuzzy logic to fault tree analysis determining the safety of basic events on basis of possibilistic distributions. Deficiencies of the classical approach are that in general the relative frequencies of the basic events are not properly known, and further they are not stationary. Hence, the tolerances of the frequency values of hazards are hardly feasible within a classical approach and the tolerances of the head effects cannot reliably be calculated. It was investigated how the possibility measure of the calculated fault event frequency depends on the assumed possibility measures of the frequencies of the basic events - mainly on those due to human factors. 

The reader should note that a fault tree includes the following (1) works backward from an undesirable event or ultimate consequence to the possible causes and failures, (2) relates the occurrence of an undesired event to one or more preceding events, (3) chain links basic events to intermediate events that are in turn connected to the top event, (4) is used in the calculation of the probability of the top event, (5) is based on the most likely or credible events that lead to a particular failure or accident, and (6) analysis includes human error as well as equipment failure. 

In this context, the top event in a fault tree can be an accident or a conflict [45]. The probabilities for accidents can come from classic accident analysis [42], the corresponding ones for conflicts or mistakes (being at the other end of the tree structure) are not generally known and hard to extract [35, 42]. An example of such calculations as well as further information, for example, on validity of the method, can be found in [45]. The method seems to be able to generate sound results, especially on the connections between conflicts and accidents, although many assumptions are basically needed during evaluation [36]. 

Once all basic event probabilities are determined for a consequence tree the risk value can be computed. The top consequence probability is obtained by the classical approach of combining the basic event probabilities according to the minimal cut set equation. However, for the mathematical combination of probabilities, dependent uncertainty analysis is used. This analysis is such that the data sources determine the coupling of uncertainties and hence the uncertainty in the risk value. In addition, a quantitative subjectivity value is calculated for the top-consequence probability estimate based on the data subjectivities of the contributing basic events. 

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