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Method 
Description,
Goal 
MTBFCalculation  Calculate
failure rates on component level. Almost always for electronic systems
according to an established calculation
standard. MTBF calculations are almost always based on BOMs. MTBF calculation is the first analysis step in the context of more sophisticated reliability and safety analyses, which use MTBF values as input data. 
FMEA / FMECA  Failure
Mode and Effect (Criticality) Analysis is used in order to determine
potential failure
modes, causes and effects. Depending on FMEA type, either calculate
probabilities of failure modes and effects, or define, track and
evaluate mitigation / improvement actions. FMEA can be performed on component level, system level, or any other intermediate level. The main goal of FMEA is to cover and address every potential failure mode. FMEA *can* be the final analysis step for simple series systems, however, for complex systems, further methods like reliability block diagram, markov or fault tree analysis may be necessary. 
Reliability Block Diagram 
System break
down on simple functional blocks. Makes sense only for complex systems with redundant paths, fault tolerant behaviour, dedicated maintenance philosophy, unique failure scenarios etc. Functional blocks are connected with arrows in order to depict a kind of "functional flow". MTBFs for the functional blocks are almost always calculated using BOMs. The focus of reliability block diagram is rather on reliability than on safety: Availability, throughput, capacity and reliability calculation on system level are the main goals. Every complete system can be described by a single Reliability Block Diagram. 
Fault
Tree Analysis 
In
contrast to reliability block diagram, Fault Tree Analysis focuses on
safety by assessing the probability of dangerous system failures. For
each dangerous system failure, a separate fault tree must be built. Fault tree analysis begins with the so called top event, which is a pricise description of a dangerous system failure. A tree is built in order to construct the failure mechanisms and system behaviour that lead to the top event. The level of detail stops with so called basic events, which are either elementary events, or such events that are not further resolved for other reasons. Fault tree analysis makes most sense for scenarios with logical dependencies 
Markov
Diagram 
Markov has
the same goals like reliability block diagram, but offers a very
different methodology. Like reliability block diagrams, markov diagrams
can have many blocks. In contrast to reliability block diagrams, each
markov block represents the complete system in a very specific state.
Markov diagrams therefore can be called state diagrams. The states are
connected with arrows representing transition rates. Markov is the preferred method for system behaviours that can not be modeled by functional blocks any more. See twin engine aircraft example. The fact that markov diagrams are based on states makes this method competitive not only with reliability block diagram, but also with fault tree Analysis: Markov analysis can handle failure probabilities as well as reliability metrics like throughput, availability, MTBF, etc. Since fault tree analysis and reliability block diagrams are easier to handle in most cases, markov is practically used only for systems where other methods fail. 
Event Tree Analysis  Event tree
analysis starts where fault tree analysis stops. Fault tree top events
are initiating events for event tree analysis. This method is used in order to analyse the effectiveness of emergency systems and fall back systems in case the undesired top event has actually happened. The focus of this method is on determining probabilities for various scenarios after the top event, or in other words, this method focuses on consequences of a dangerously failed system. 
Weibull Analysis  Although
very often based on the weibull distribution, this method is not
limited to the weibull distribution. Weibull analysis is used in order to determine MTBF, lifetime, and shape of the lifetime distribution (which is very often a weibull distribution) from experimental data like field data and laboratory test data. The weibull distribution (as well as other eligible distribution functions) with its two parameters has proven to be a good means for modeling failure rates over time, however, with very limited mathematical foundation, it just works well in practice. By evaluating MTBF and lifetime, the goal of this method is similar to MTBF calculation based on BOMs. However, the quality of the weibull analysis results as well as their certainty are way better than MTBF calculation results. Unfortunately, effort (time and money) needed for weibull analysis is way higher than for MTBF calculation based on BOMs 
Accelerated
Tests 
Like weibull
analysis, accelerated tests use weibull and other eligible distribution
functions in order to determine MTBF, lifetime and shape parameter from
laboratory test data, and therefore has the same primary goal like
weibull analysis. The secondary goal of accelerated tests is to establish a test plan that ensures minimum test effort while maximizing the certainty of the test results. In particular: How many units have to be put on test, which test time is needed and at which temperatures should the test be run. Dedicated statistical methods are available in order to determine the optimal test strategy. Like weibull analysis, accelerated tests are used in order to determine MTBF, lifetime and shape parameter. Also like weibull analysis, accelerated tests need way more effort than MTBF calculations based on BOMs. 
Further Methods  Depending on
the nature of the problem, special mathematical methods may be needed. Examples: Error rates in signal transmission and software safety. 