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The following methods are common in quantitative reliability analysis and in functional safety analysis.

Description / Goal
RAMS is not a method, but an abbreviation for reliability, availability, maintainability and safety.
Here is a list of standards covering RAMS topics.
MTBF Calculation
MTBF is a basic and intuitive reliability metric. MTBF calculation is often the first step of a reliability or safety analyses. In most cases, especially in functional safety, the analysis subject is an electronic system, whose MTBF is calculated according to a worldwide recognized MTBF standard.
FMEA/FMECA Failure Mode and Effect (Criticality) Analysis is used in order to determine potential failure modes, causes and effects. Depending on requirements and goals, FMEA can be tailored accordingly, for example:
  • Calculation of probabilities of failure modes and effects (quantitative FMEA)
  • Evaluation and mitigation of defects, planning and tracking of improvement actions (qualitative FMEA).
FMEA can be performed on piece part level, system level, or any other intermediate level. 
FMEA *may* be the final analysis step for simple (= neither redundant nor fault tolerant) systems, however, for complex systems, further methods like reliability block diagram, markov or fault tree analysis may be neccessary.

Makes sense only for complex systems with redundant paths, fault tolerant behaviour, dedicated maintenance philosophy, unique failure scenarios etc.
The system under consideration is broken down into functional blocks.
Functional blocks then 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.
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. So called basic events, which are either elementary events, or events that are not further resolved for other reasons, are on the lowest level of a fault tree. 
Fault tree analysis makes most sense for scenarios with functional dependencies (AND-type connections of events), but for some reason is also sometimes used for simple systems,  where it is actually useless, in particvular when an FMEA already exists.
Markov Diagram Markov has the same goal like reliability block diagram, but offers a very different methodology. Like reliability block diagrams, markov diagrams can have many blocks. But in contrast to reliability block diagrams, markov blocks don't  represent functional elements, but the complete system in a specific state. Markov diagrams therefore can be called state diagrams. The states are connected with arrows representing transition rates between states. 
Markov is the preferred method for systems whose behaviour can not be sufficiently modeled with functional blocks. The twin engine aircraft example explains this in detail.
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 underrstand, 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 used as 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 (of a fault tree) has actually occurred. 
The focus of this method is determining probabilities for various scenarios after the top event, or in other words, this method focuses on consequences of dangerously failed systems.
Weibull Analysis Although 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 field failure data or laboratory test data.
The weibull distribution (as well as other distribution functions) with its two parameters has proven to be a good means for modeling failure rates over time, however, with actually no mathematical foundation;  the reason why it is used is because it just works well in practice. 
By evaluating MTBF and lifetime, the goal of weibull analysis is similar to MTBF calculation based on BOMs. However, if the quality of the data is not too bad, the quality of the results can be way better than MTBF calculation results with established MTBF standards
Unfortunately, effort (time and money) needed for weibull analysis is usually higher than for MTBF calculation based on BOMs using established MTBF calculation standards.

Accelerated Tests Like weibull analysis, accelerated tests use weibull or other eligible distribution functions in order to determine MTBF, lifetime and shape parameter from laboratory test data, and therefore have 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 using established MTBF calculation standards.

Further Methods Depending on the nature of the problem, special mathematical methods may be appropriate. 
Example: Error rates in signal transmission and software safety.

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