# FailureDistribution

FailureDistribution[bexpr,{{x1,dist1},{x2,dist2},}]

represents the failure distribution for a system with events xi having reliability distribution disti where the top event occurs when the Boolean expression bexpr is True and event xi has occurred when xi is True.

# Details

• FailureDistribution[bexpr,] corresponds to a fault tree specification.
• The Boolean expression bexpr is also known as the structure function for the system.
• Typical structure functions include:
•  Or gate And gate -out-of- system consecutive--out-of- system
• The structure function bexpr can be any positive unate Boolean function.
• UnateQ[bexpr] can be used to test whether a Boolean expression is positive unate.
• The event reliability distributions disti need to be univariate with PDF[disti,t] zero for t0.
• For FailureDistribution[bexpr,] with event indicators xi:
•  xiTrue indicates event xi has occurred xiFalse indicates event xi has not occurred
• The CDF at time t for FailureDistribution[bexpr,{{x1,dist1},}] is given by Probability[bexpr/.{x1->t1t,},{t1dist1,}].
• FailureDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, and RandomVariate.

# Examples

open allclose all

## Basic Examples(3)

Define a system whose top event occurs if either of the two events occurs:

Compute the survival function:

Define a system whose top event occurs if both of the underlying events occur:

Compute the survival function:

A structure with both an And and an Or gate:

Distribution functions:

Mean and median time to failure:

The probability that the system fails before time :

## Scope(22)

### Basic Uses(5)

Find the mean time to failure for a system that fails if either of the two events occur:

Find the survival function for a system that fails if both events occur:

Find the SurvivalFunction of a system that fails when two out of three events occur:

Generate random numbers for a system with different event distributions:

Compare the histogram to the PDF:

Model a system with a component that has standby functionality:

Compute the mean time to failure:

### Structure Functions(4)

A system with four events:

A system with three events and a voting gate:

A system that fails if at least three out of four events occur:

Any positive unate Boolean expression can used as a structure function:

Use UnateQ to test whether a Boolean expression is positive unate:

Use any parametric lifetime distribution including LogNormalDistribution:

Numerically compute the mean time to failure:

Compute the mean time to failure as a function of distribution parameters:

When one event occurs substantially later, it determines the system lifetime:

Define two distributions with the same mean time to failure:

Define systems with Or and And gates with these distributions:

Show how the different mean time to failures vary with :

Discrete lifetime distributions can be used:

Calculate the mean time to failure:

Plot the SurvivalFunction:

Model airplane glass strength using SmoothKernelDistribution:

Consider a system that fails if either the airplane glass breaks or the other event occurs:

Model events with HistogramDistribution:

Plot the survival function for an Or gate with two and three events:

Model an event directly from data with EmpiricalDistribution:

Plot the survival function when either of two events causes the system to fail:

Model events with StandbyDistribution:

Calculate the mean time to failure:

Plot SurvivalFunction:

Complex systems can be modeled in steps:

Find the mean time to failure:

This is equivalent to modeling the whole system at once:

Model an event with StandbyDistribution:

The first of two events to occur causes the top event to occur:

Plot the survival function and compare it with only the standby event:

A system where one event is ParameterMixtureDistribution:

Model a two-component cold standby system with MixtureDistribution:

Use it in FailureDistribution and compute the survival function:

Show equivalence with StandbyDistribution:

OrderDistribution can be used to model lifetimes of events:

Plot SurvivalFunction:

## Applications(6)

Model the risk of not being woken up in the morning. Assume that an old alarm clock is kept as a backup:

The main alarm clock is electrical:

Define the distributions; the rates are failures per year:

Find the probability of not being woken up at some point during the first year:

Mean time to failure in years:

A problem at coal mines is bulldozers falling through bridged voids in coal piles. The bulldozer can be over a void intentionally or unintentionally:

To form a void, there has to be subsurface flow in the coal. This requires removal of coal from below on a conveyor belt, and an open feeder to that belt:

It is also required that no flow occurs on the surface. This can happen if the coal freezes:

Compacted coal can also lead to a non-flowing surface:

The complete fault tree:

Assume the following distributions for the events:

The probability of the main event happening during a year on one coal pile:

For the 337 piles in the United States:

The mean time to the event happening on one pile is roughly 60 years:

An underwater dry maintenance cabin is used to repair pipelines underwater. The mean time to failure in hours for the life support system components and their life distributions are given below:

First model the air supply subsystem:

The exhaust system:

Finally, model the air detection system:

The life support system fails if any of the subsystems fail:

The typical mission time is 24 hours. Compute the probability of survival for a mission:

The mean time to failure is considerably larger than the mission time:

Consider a propulsion system that provides thrust to a vehicle in orbit around the Earth. We model the event of applying thrust after the device has been turned off:

Model the emergency switch:

Failure of the timing relay to open:

The structure for failure of relief valve 1 to close:

The structure for failure of relief valve 2 to close:

The unwanted propulsion occurs if both relief valves fail to close:

Compute the mean time to an unwanted propulsion:

Find the probability of unwanted propulsion occurring during a mission that lasts six months:

A system has the following structure and lifetime distribution:

Find the requirement on the failure rate to avoid failure during a mission of five days with a probability of :

FailureDistribution can be used as a generalized OrderDistribution:

This is equal to OrderDistribution if identical distributions are used:

## Properties & Relations(12)

FailureDistribution uses local names for variables in the input:

Hence, subsequent computations can be done with the original variable name:

The probability that neither of two failure events occur before :

This corresponds to an Or gate:

The probability that both failure events did not occur before :

This corresponds to an And gate:

The probability that no two failure events occur before :

This corresponds to a two-out-of-three voting gate:

An Or gate connecting identical events corresponds to an OrderDistribution:

An And gate connecting identical events corresponds to an OrderDistribution:

A voting gate with identical events corresponds to an OrderDistribution:

The lifetime of Or gate-connected basic events is the minimum of the component lifetimes:

The lifetime of two events connected with an And gate is the maximum of the event lifetimes:

A -out-of- voting gate corresponds to TransformedDistribution with a RankedMin function:

Compare the survival functions:

Exponentially distributed events connected by an Or gate give an exponential top event:

Weibull distributed events connected by an Or gate give a Weibull distributed top event:

FailureDistribution models that if the failure of or occurs, the top event occurs:

ReliabilityDistribution models that both components have to work for the system to work:

Components and working is equivalent to the event of either or failing:

Model that two out of four components need to fail for the system to fail:

This is equal to that three out of four components need to work for the system to work:

## Possible Issues(3)

Component distributions need to have a positive domain:

Use TruncatedDistribution to restrict the domain to positive values only:

Exact or symbolic properties cannot always be computed:

Approximate values can typically still be found:

A FailureDistribution is only well defined for positive unate structure expressions:

Use UnateQ to test whether a Boolean expression is positive unate:

## Neat Examples(1)

To get from one point to another, you have the alternatives of riding a horse, driving a car, driving a tank, or taking the boat. The boat also requires that no sharks are around. Find the mean time to failure:

Wolfram Research (2012), FailureDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FailureDistribution.html.

#### Text

Wolfram Research (2012), FailureDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FailureDistribution.html.

#### CMS

Wolfram Language. 2012. "FailureDistribution." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FailureDistribution.html.

#### APA

Wolfram Language. (2012). FailureDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FailureDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_failuredistribution, author="Wolfram Research", title="{FailureDistribution}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/FailureDistribution.html}", note=[Accessed: 22-July-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_failuredistribution, organization={Wolfram Research}, title={FailureDistribution}, year={2012}, url={https://reference.wolfram.com/language/ref/FailureDistribution.html}, note=[Accessed: 22-July-2024 ]}