BASIC CONCEPTS AND AN OVERVIEW - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 117

Early use of Monte Carlo techniques was made for the quantitative evalua-
tion of fault trees. While some effort has continued in the use of purely Monte
Carlo methods, they have largely been supplanted deterministic techniques often
referred to as Kinetic Tree methods. Two limitations, however, present themselves
in the use of Kinetic Tree methods. First, in Kinetic Tree methods the reliability
characteristics of each component are modeled separately. To evaluate the fault
tree by combining component failure probabilities, the components are assumed
to have independently of one another. In fact, dependencies often arise from com-
mon mode failures, from the increased stress in partially disabled systems, and
from a variety of errors in testing, maintenance and repair. Due to this limitation
of the Kinetic Tree formulation there is increasing use of Markov models for reli-
ability analysis, for with such models quite general dependencies between com-
ponents may be treated. For systems with more than a few components, however,
Markov analysis by deterministic means becomes a prodigious task. For even
while innovative methods have been employed to reduce the complexity of the
computations, the fact remains that one must solve a set of 21 coupled first-order
differential equations, thus even a system with only ten components will result
in a system of over one thousand coupled equations with a transition matrix with
over a million elements. Moreover, if some of the components are repairable,
the equations are likely to be quite stiff, requiring that very small time steps be
used in the numerical integration. A second limitation on Kinetic Tree methods
is a result of the lack of precision to which the component failure and repair rates
are normally known. Invariably this is accomplished by Monte Carlo sampling
of the failure rate data using log-normal or other distributions. The fault tree is
evaluated deterministically with data from each data sampling, and the mean,
variance and other characteristics of the system are estimated. A similar proce-
dure is also applied to Markov models, requiring that the solution of the coupled
set of differential equations be repeated thousands of times. What follows is the
formulation of a class of Monte Carlo methods, which provides a natural frame-
work for the treatment of both component dependencies and data uncertainties.
Some researchers formulate Monte Carlo simulation of the unreliability of sys-
tems with repairable components within the framework of a Markov process. This
approach retains the power of deterministic Markov methods in modeling com-
ponent dependencies that would not be possible if direct Monte Carlo simulation
were to be carried out. At the same time the Monte Carlo simulation requires very
little computer memory. Variance reduction techniques, similar to those that have
been highly developed for neutral particle transport calculations, are applied to
greatly increase the computational efficiency of Monte Carlo reliability calcula-
tions. Monte Carlo formulation is generalized to include probability distributions
that represent the uncertainty in the component failure and repair rate data. The
variance in the result is then due to two causes: the finite number of random walk
simulations, and the uncertainty in the data. A batching technique is introduced