Stochastic failure modeling

Whiteside, M B., et al. 2012. Stochastic failure modelling of unidirectional composite ply failure. Reliability Engineering and System Strfety 108(0) 1-9. 

Due to the dynamic behavior of reconfigurable fault-tolerant systems, the creation of stochastic dependability models is a difficult task. Traditional techniques like fault trees or rehabdity block diagrams are no longer sufficient in many cases, because they assume all components to be of a Boolean nature. However, in today s adaptable and reconfigurable systems, components must be described by more than the states active and failed in order to reflect the different roles of a component in a reconfigurable system. Moreover, often the system itself is not considered to be Boolean, but different failure classes are discriminated. Finally, the basic events (component failures and repairs) cannot be assumed to be independent, but common cause failure, failure propagation, limited repair capacities etc. must be taken into account. 

Consider a continnous-time model with stochastic failure times. The following notation is used. For k >1 the stochastic variable T represents the interval between the (k-l)th and the kth failure event. Using ... 

The time-dependent evolution of the system and process dynamics in interaction with the stochastic failure behaviour of safety systems and hiunan actions is reduced to static cause-effect models which operate with fixed probabilistic assessments for the stochastic behaviour. The order of events is predetermined by the expert and may possibly represent the chronological order of some reference sequences, but the question is, whether it is apphcable to all sequences. What is the consequence, if specific process conditions induce another order of different events ... 

Event-tree models in a Level 1 PSA generally account for the order of demands of safety system functions at set points and for the (stochastic) failure behaviour of the required functions. It is common practice to consider just two alternative states at each set point, namely required function is successful and required function fails . No satisfactory consideration is given, for instance, to situations where technical safety systems which are successfully started fail to function with the required capacity and / or fail to run within the required mission time. What is the consequence, if sequences accounting for stochastic failure times of safety system functions are not considered Is the resulting spectrum of event sequences still sufficient enough to obtain an adequate probabilistic assessment for (core) damage states How reahstic are the probabilistic assessments for damage states derived from static event tree models ... 

For some aspects of model uncertainties, well-known quantification methods are available. A Bayesian approach might be practicable, for instance, to quantify the uncertainty on the probability model to apply for the stochastic failure behaviour of system components. Monte Carlo analysis might be appropriate to quantify the uncertainty resulting from the application of thermal-hydraulics codes. 

Four degraded components of the wind turbine are considered. The failure models are based on onshore ones and derived using an empirical approach based on stress factors for mechanical systems. Generalized Stochastic Petri Nets (GSPN)... 

Standard procedures that are used for testing of construction materials are based on square pulse actions or their various combinations. For example, small cyclic loads are used for forecast of durability and failure of materials. It is possible to apply analytical description of various types of loads as IN actions in time and frequency domains and use them as analytical deterministic models. Noise N(t) action as a rule is represented by stochastic model. 

Bedaux and Kooijman 1994 Kooijman 1996 Newman and McCloskey 1996, 2000 Zhao and Newman 2007). This is not just an academic discussion the 2 theories lead to different time courses of mortality at constant exposure (Kooijman 1996) (see Figure 2.10) and have very different consequences for sequential exposure (Newman and McCloskey 2000 Zhao and Newman 2007). In reality, both sensitivity difference and stochasticity are likely to play a role in mortality. Individuals also differ in sensitivity, especially in field populations, but there is clearly a substantial stochastic component involved in mortality that cannot be ignored. The method to deal with stochastic events in time is survival analysis or time-to-event analysis (see Bedaux and Kooijman 1994 Newman and McCloskey 1996). For industrial practices, this method has a long history as failure time analysis (see, e.g., Muenchow 1986). Bedaux and Kooijman (1994) link survival analysis to a TK model to describe survival as a function of time (i.e., the hazard rate is taken proportional to the concentration above a threshold value). Newman and McCloskey (1996) take an empirical relationship between external concentration and hazard rate. 

The first example is due to Rudd (1960) and concerns the optimal number of replications of the preparation of chemicals needed in a main process. If these preparations are subject to failure and if it is quite essential that they should be available without delay to the main process, it will be advisable to prepare more than one batch but this will become expensive if too many are prepared. The second example is a transliteration into manufacturing terms of Bellman s stochastic gold mining process (1957, Chapter 2). The third is a consideration of some of the simpler models of catalyst replacement policy that have been discussed by Roberts (1960a,b). 

Simulation All business processes have random components. Sales may take one value or another. A machine may or may not fail. Often these random, or stochastic, elements of a problem make analyzing it very difficult. In these cases, simulation is often an effective tool to help with decisions. In simulation, a model of the process is created on a computer. Each of the random elements of the model (sales, failures, etc.) is specified with a probability distribution. When the model is run, the computer simulates carrying out the process. Each time a random event occurs, the computer uses the specified probability distribution to randomly decide what happens. 

In many ways modeling the repair process is difficult because the repair process is quite different from the failure process. Random failures are due to a stochastic process and most of our modeling techniques were created for these stochastic processes. Certain aspects of the repair process are deterministic. Other aspects of the repair process are stochastic. Fortunately, we can approximate the repair process more accurately with Markov models than most other techniques. 

A reliability model. The sequence of failures is modeled as a stochastic process. This model specifies the failure behavior process. The model parameters are determined by fitting a curve to failure data. This implies also a need for an inference procedure to fit the curve to data. The reliability model can then be used to estimate or predict the reliability (see Section IV). 

Failure count models are based on the number of failures that occur in different time intervals. The number of failures that occur is, with this type of model, modeled as a stochastic process, where N(t) denotes the number of failures that have occurred at time t. 

Jiang R, Murthy DNP 1998. Mixtures of Weibull distributions parametric characterization of failure rate liinction. Applied Stochastic Models and Data Analysis 14, 47 5. 

Figure 2 shows an ECSPN version of the anonymous PN model from Fig. 1, the system RBD model in the upper right comer and the ECSPN component model below. In the upper left comer the declarations of the ECSPN model can be seen. The token within the model contains the pre-age of the component which is considered when calculating the component s stochastic lifetime. This can be seen by the depiction of the arc inscription AgeEn = a. The age of a component is equivalent to its instantaneous failure probability. The... 

We characterize a Monte-Carlo simulation ofthe models by the triplet (a, p, y) representing respectively, the failure probabihty, the cost intensity, and the responsiveness parameter. One simulation is obtained by running the stochastic models a large number of times, yielding different state trajectories at each run. An... 

All of those points are very hard to model within FTA (Fault Tree Analysis). An occurrence of lethal combination of input parameters (1. a) is a stochastic process and depends on the frequency of initiation. It is not possible to disclose it by periodical test and its failure rate is decreasing in time. It is hke maturing instead of aging. 

Fricks, R. and K. Trivedi (1997). Modeling failure dependencies in reliability analysis using stochastic Petri nets. In Proceedings of the 11th European Simulation Multiconference Istanbul, Turkey, June 1—4, 1997. 

They most often lead to exponential, eventually WeibuU probabilities distribution. The time curves of fault rate and reparations, eventually other stochastic influences during the reliabUily of complicated systems ensuring in real operation, are not taken into account in these models (CHOVANEC, A.). In the real case the operation reliability, respectively its partial properties are coimected with processes, which are necessary for the failure removal (control process, supplying system, repairing process, etc.). That s why also the model may have several states and distrihutions of random variables. 

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