تجزیه و تحلیل حساسیت احتمالی دسترس پذیری سیستم با استفاده از فرایندهای گاوسی

The availability of a system under a given failure/repair process is a function of time which can be determined through a set of integral equations and usually calculated numerically. We focus here on the issue of carrying out sensitivity analysis of availability to determine the influence of the input parameters. The main purpose is to study the sensitivity of the system availability with respect to the changes in the main parameters. In the simplest case that the failure repair process is (continuous time/discrete state) Markovian, explicit formulae are well known. Unfortunately, in more general cases availability is often a complicated function of the parameters without closed form solution. Thus, the computation of sensitivity measures would be time-consuming or even infeasible. In this paper, we show how Sobol and other related sensitivity measures can be cheaply computed to measure how changes in the model inputs (failure/repair times) influence the outputs (availability measure). We use a Bayesian framework, called the Bayesian analysis of computer code output (BACCO) which is based on using the Gaussian process as an emulator (i.e., an approximation) of complex models/functions. This approach allows effective sensitivity analysis to be achieved by using far smaller numbers of model runs than other methods. The emulator-based sensitivity measure is used to examine the influence of the failure and repair densities' parameters on the system availability. We discuss how to apply the methods practically in the reliability context, considering in particular the selection of parameters and prior distributions and how we can ensure these may be considered independent—one of the key assumptions of the Sobol approach. The method is illustrated on several examples, and we discuss the further implications of the technique for reliability and maintenance analysis.

In this paper, we present a new approach to study the sensitivity analysis of availability. In general sensitivity analysis is concerned with understanding how changes in the model input (distribution parameters) would influence the output. Suppose that our deterministic model can be written as y=f(x)y=f(x), where xx is a vector of input variables (or parameters) and y is the model output. For example, the inputs could be considered as the parameters of the failure and repair densities, θθ, and the output could be the availability A(t,θ)A(t,θ) at time t. The traditional method of examining sensitivity of a model with respect to the changes in its input variables is local sensitivity analysis which is based on derivatives of f(·)f(·) evaluated at some ‘base-line’ (or central estimate) x=x0x=x0 and indicates how the output y will change if the base line input values are slightly perturbed (see [11] for the different local sensitivity measures commonly used in Bayesian analysis). This is clearly of limited value in understanding the consequences of real uncertainty about the inputs, which would in practice require more than infinitesimal changes in the inputs. Furthermore, these methods are computationally very expensive for complex models and usually require a considerable number of model runs if we use a Monte Carlo based method to compute these sensitivity measures. For instance, Marseguerra et al. [21] used Monte Carlo simulation to calculate the first-order differential sensitivity indexes of the basic events characterising the reliability behaviour of a nuclear safety system. They reported that the computation of the sensitivity indexes for the system unavailability at the mission time by Monte Carlo simulation requires 107107 iterations. In another study, Reedijk [28] reported that first order reliability methods and Monte Carlo simulation have certain disadvantages and some problems could not be solved with these methods. This issue is particularly interesting in the case where the model is computationally expensive so that simply computing the output for any given set of input values is a non-trivial task. This is especially the case for large process models in engineering, environmental science, reliability analysis, etc. that may be implemented in complex computer codes requiring many minutes, hours or even days for a single run. However, in order to implement many of the standard sensitivity techniques discussed by [30] we require a very large number of model runs. In that case even for a model that takes just one second to run, many sensitivity analysis measures may take too long to compute. The most frequently used sensitivity indices are due to Sobol [32]. However, these require an assumption of independence (as discussed by Bedford [1]). Hence in Section 3 we discuss here how one might go about choosing an appropriate parameterisation in which the sensitivity analysis can be carried out using independent uncertainty variables. It should be noticed that the probabilistic sensitivity analyses are often effectively carried out with efficient sampling procedures (e.g., [14] and [15]), but these procedures are computationally very expensive. Therefore, we present an alternative computational tool to implement sensitivity analysis based on the work of [22]. This is a Bayesian approach of sensitivity analysis which unifies the various methods of probabilistic sensitivity analysis which will be briefly introduced in Section 2. This approach is computationally highly efficient and allows effective sensitivity analysis to be achieved by using very smaller numbers of model runs than Monte Carlo methods require. The range of tools used in this approach also enables us to do uncertainty analysis, prediction, optimisation and calibration. Section 4 presents this method. This paper extends work carried out by Daneshkhah and Bedford [10] where emulators were used to examine the influence of failure and repair densities' parameters on the system availability of a simple one component repairable system where Exponential and Weibull were considered as distributions for the failure and repair rates. Here, we consider the sensitivity analysis of repairable systems with more than one component, chosen so that we can use numerical integration to compute availability. The systems are: a parallel system with two components where the failure and repair distributions are exponentials; a well known standby-redundancy system where there are three parameters to be examined and the failure and repair distributions are also exponentials; and move-drive system with eight components and 17 parameters, where the repair rate is constant but the failure distributions are Weibulls. There are closed forms for the availability functions for the first two systems and we use them to validate the method. But there is no closed form for the third system, and to evaluate the system availability at the selected parameter values, an expensive numerical method required. We present some conclusions and possible future developments in Section 6.