# ReliabilityDistribution

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

represents the reliability distribution for a system with components xi having reliability distribution disti, where the whole system is working when the Boolean expression bexpr is True, and component xi is working when xi is True.

# Details • ReliabilityDistribution[bexpr,] corresponds to a reliability block diagram specification.
• The Boolean expression bexpr is also known as the structure function for the system.
• Typical structure functions include:
• series system parallel system k-out-of-n system consecutive-k-out-of-n 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 component reliability distributions disti need to be univariate with PDF[disti,t] zero for t0.
• For ReliabilityDistribution[bexpr,] with component indicator variables xi:
•  xiTrue indicates component xi working xiFalse indicates component xi failed
• The survival function at time t for ReliabilityDistribution[bexpr,{{x1,dist1},}] is given by Probability[bexpr/.{x1->t1>t,},{t1disti,}].
• ReliabilityDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, and RandomVariate.

# Examples

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## Basic Examples(3)

A series-connected system:

Mean time to failure:

A parallel-connected system:

Mean time to failure:

One component in series with two parallel components:

Distribution functions:

Mean and median time to failure:

The probability that the system fails before time :

## Scope(26)

### Basic Uses(4)

Find the mean time to failure for a parallel system with two components:

Find the SurvivalFunction of a system that needs two out of three components to work:

Find the mean time to failure (MTTF) for a serial system with one standby component:

Generate random numbers for a serially connected system:

Compare the histogram to the PDF:

### Structure Functions(7)

A system with two components in series with two in parallel:

A system with three components and a voting gate:

A system that works if at least three out of four components work:

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

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

Construct a multi-stage system, where each component is a -out-of- system:

A redundant consecutive -out-of- with redundancy in each consecutive structure component:

Series of consecutive -out-of- systems:

A system with an increasing and decreasing failure rate:

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 component has a substantially longer lifetime, it determines the system lifetime:

Define two distributions with the same mean time to failure:

Define a parallel and serial system with these distributions:

Show how the different mean times to failure 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 series-connected system with one component being airplane glass:

Model components with a HistogramDistribution:

Plot the survival function for a serial system with two and three components:

Model components directly from data with EmpiricalDistribution:

Plot the survival function for a parallel system with two components:

Model a component with a StandbyDistribution:

Calculate the mean time to failure:

Plot the 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 a component with a StandbyDistribution:

Both components need to work for the system to work:

Plot the survival function and compare with only the standby component:

A system with a ParameterMixtureDistribution:

Model a two-component cold standby system with a MixtureDistribution:

Use it in ReliabilityDistribution and compute the survival function:

Show equivalence with StandbyDistribution:

OrderDistribution can be used to model component lifetimes:

Plot the SurvivalFunction:

MarginalDistribution can be used to model component lifetimes:

Plot the SurvivalFunction:

## Applications(12)

Model the launch of an aircraft. The hangar door can be opened electronically or manually:

Two fuel pumps require power to run:

Two more pumps run on reliable batteries, giving the following fuel transfer structure:

Also needed is deicing of the aircraft and a fuel storage tank:

Define distributions with failure rates per hour. Let the failure rate of the power supply be variable:

Compute the survival function and the mean:

Study how the reliability varies when the failure rate is decreased from /hour:

A vacuum system in a small electron accelerator contains 20 vacuum bulbs arranged in a circle. The vacuum system fails if at least three adjacent vacuum bulbs fail:

Plot the survival function:

Compute the mean time to failure:

A satellite needs power from its battery, which is recharged by solar panels. Calculate the number of load cycles the system can stand:

The battery has a standby battery:

Simulate the number of load cycles the satellite can manage:

Consider a skateboard with a ball bearing in each of its wheels. Find out the lifetime of such a skateboard:

Model the ball bearings with a known distribution:

Assume the skateboard can be used when at least three out of four wheels are functioning:

Plot the survival function:

A car has four tires that are all needed to drive. Given the lifetime distribution of a tire as ExponentialDistribution[0.0004], calculate the failure rate and mean time to failure (MTTF) of the car with respect to the tires:

Simulate the lifetime of the car:

In order to fly, an airplane needs both wings, one engine out of two on each wing, and the air control system operating. Model the engine system:

Model the air control system:

Put it all together:

Calculate the mean time to failure for the airplane:

Calculate the probability that a six-hour flight will be successful:

Find the probability that an airplane working after 300 hours still works after an additional 200 hours:

Plot the survival function:

Simulate the lifetime of the airplane:

A data center needs 20 servers to work, one power supply, one cooling system, and one network connection. The power supply has a standby component. The network connection consists of two parallel connections. Model the server:

Model the power supply with StandbyDistribution:

Model the data center:

Calculate the mean time to failure (MTTF):

Simulate the uptime of the data center:

Find the component that is most important to improve in a single server:

A system has the following structure and lifetime distribution:

Find the requirement on the failure rate to survive a mission of five days with a probability of 0.9995: A solar panel consists of arrays of photovoltaic cells. Each array has 10 cells:

The solar panel requires three out of five arrays and an inverter to work:

Find the mean time to failure in years:

Simulate the lifetimes of solar panels:

A company that sells solar panels gives a 10-year warranty. An additional 10 years can be added. Find out how the company should price this extended warranty:

A logistics company uses trucks, a ship, and a train to deliver goods to customers. Two out of three trucks and clear roads are needed for road transport. A working ship and calm weather conditions are needed for the ship, and a functioning train and tracks are needed for the train. Model the road transport:

Model the sea transport:

Model the train transport:

Put the system together:

Calculate the expected number of days before the system fails to deliver to the customers:

The company is considering leasing an airplane to improve the reliability of delivery. Calculate how much this airplane would improve the mean time to failure:

Compare the two systems:

Assume that Charles Lindbergh had the option to select a one-engine or a two-engine aircraft for his flight between New York and Paris. Investigate which configuration of engines gives the highest reliability:

Plot the survival functions for these different configurations:

It is clear that the best choice was two engines in parallel, which at the time was not available, and Lindbergh therefore chose one engine. Calculate the probability that one engine survives a 33.5-hour flight:

Model a launch into space in three phases: a ground check, a launch phase, and in orbit. Define the lifetime distributions for operating and inactive modes:

For the ground phase, an environmental accelerator of 5 is used:

The launch phase presents a harsher environment, with the environmental factor 400:

The environment in orbit is not very harsh, so a factor 1 is used:

If the ground check takes 24 hours, the launch 0.1 hours, and the orbit phase 100 days, find the probability of success:

## Properties & Relations(14)

ReliabilityDistribution uses local names for variables in the input:

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

The probability that two components both have a lifetime greater than :

This corresponds to a series connection:

The probability that at least one component has a lifetime greater than :

This corresponds to a parallel connection:

The probability that at least two components have a lifetime greater than :

This corresponds to a 2-out-of-3 system:

A series connection of identical components corresponds to an OrderDistribution:

A parallel connection of identical components corresponds to an OrderDistribution:

The lifetime of a series connected system is the minimum of the component lifetimes:

The lifetime of a parallel connected system is the maximum of the component lifetimes:

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

Compare the survival functions:

A series system of exponentially distributed components is exponential:

A series system of components with a Weibull lifetime distribution is another Weibull:

ReliabilityDistribution models that component or has to work for the system to work:

The corresponding FailureDistribution models the events that both and fail:

Component or working is equivalent to the event of both and failing:

Model two out of four components needing to work for the system to work:

This is equal to three out of four components needing to fail for the system to fail:

ReliabilityDistribution[f[x1,],]FailureDistribution[¬f[¬x1,],]:

Retrieve the ReliabilityDistribution for a SystemModel with reliability information:

## 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 ReliabilityDistribution is only well defined for positive unate structure expressions: Use UnateQ to test whether a Boolean expression is positive unate:

## Neat Examples(2)

Show the hazard function for all systems with up to four identical components:

Generate systems:

Build list of unique hazard functions:

Plot hazard functions and corresponding systems:

Use a Graph as a reliability block diagram:

Calculate the mean time to failure with the green and red vertices as the start and end vertices: