# StandbyDistribution

StandbyDistribution[dist1,{dist2,,distn}]

represents a standby distribution with component lifetime distributions disti. When component fails, component will become active.

StandbyDistribution[dist1,{dist2,,distn},p]

represents a standby distribution where switching from component to component succeeds with probability p.

StandbyDistribution[dist1,{dist2,,distn},sdist]

represents a standby distribution where the switch component has lifetime distribution sdist.

StandbyDistribution[dist1,{,{disti,inactive,disti,active},},]

represents a standby distribution where the  component lifetime distribution follows disti,inactive in inactive mode and disti,active in active mode.

# Details  • StandbyDistribution[,] represents a system with perfect switching where transitioning between components always succeeds.
• StandbyDistribution[,,s] represents a system with imperfect switching. If s is a distribution, it represents that lifetime of the switch; otherwise it represents the probability of a successful transition between components.
• StandbyDistribution[,{,Ai,},] represents a standby distribution where the  component follows a cold standby distribution Ai when it is active, and does not deteriorate when it is inactive.
• StandbyDistribution[,{,{Ii,Ai},},] represents a standby distribution where the  component follows a warm standby distribution. The component deteriorates following distribution Ii when it is inactive and distribution Ai when it is active.
• Any mix of cold and warm standby component distributions can be used.
• The survival function and other properties for StandbyDistribution can be derived from the equivalent TransformedDistribution[expr,dists] with the distribution assumptions dists given by {a1A1,a2A2,,i2I2,i3I3,,sS,uUniformDistribution[{0,1}]}.
•  StandbyDistribution[…] TransformedDistribution[…,dists] a1+a2+a3+⋯ A1,{A2,A3,…},p a1+ a2Boole[p>u]+a3Boole[p2>u]+⋯ A1,{A2,A3,…},S a1+a2Boole[s>a1]+a3Boole[s>a1+a2]+⋯ A1,{{I2,A2},{I3,A3},…} a1+a2Boole[i2>a1]+a3Boole[i3>a1+a2Boole[i2>a1]]+⋯ A1,{{I2,A2},{I3,A3},…},p a1+a2 Boole[i2>a1∧p>u]+a3Boole[i3>a1+ a2Boole[i2>a1]∧p2>u]+⋯ A1,{{I2,A2},{I3,A3},…},S a1+a2 Boole[i2>a1∧s>a1]+a3Boole[i3>a1+a2Boole[i2>a1]∧s>a1+a2Boole[i2>a1]]+⋯
• StandbyDistribution can be used with such functions as Mean, SurvivalFunction, HazardFunction, ReliabilityDistribution, and RandomVariate.

# Examples

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## Basic Examples(3)

Define a cold standby system with perfect switching:

Compute its PDF:

Mean time to failure:

Compare to a non-standby system:

Define a cold standby system with imperfect switching:

Compute its PDF:

Mean time to failure:

Compare to a non-standby system:

Define cold and warm standby systems, with inactive failure rate half the active failure rate:

Compute the mean time to failure:

Compare the survival functions:

## Scope(17)

### Cold Standby and Perfect Switching(3)

Define a cold standby system with three identical components:

Compute mean time to failure:

Define a cold standby system with two different components:

Compute the survival function:

Study a component with three identical components in standby:

Generate random variates:

Compare with the probability density function:

### Cold Standby and Imperfect Switching(4)

A cold standby system where the switch succeeds with probability p:

Find the mean time to failure:

Compare perfect switching to imperfect switching where the switch works half the time:

A cold standby system where the switch is modeled by a lifetime distribution:

Find the survival function:

Study the effect of having worse switches:

Cold standby system with three components and a switch modeled by a distribution:

Generate random variates:

Compare with the probability density function:

A switch modeled with a probability of success:

Compare with the probability density:

### Warm Standby and Perfect Switching(3)

Standby system where the component can fail while in standby:

Find the mean time to failure:

System with multiple components that can fail in standby:

Compare the survival function to a cold standby system:

Warm standby system with two components in standby:

Generate random variates:

Compare with the probability density function:

### Warm Standby and Imperfect Switching(4)

Warm standby system where the switch succeeds with a probability p:

Compute the mean time to failure:

Warm standby system where the switch has a lifetime distribution:

Compute the mean time to failure:

Warm standby system where the switch is modeled with a lifetime distribution:

Generate random variates:

Compare with the probability density function:

System where the switch succeeds with a probability:

### Mixed Warm and Cold Standby Systems(3)

Standby system where the second component can fail while in standby:

The system where the second and third component switch places:

Compare the survival functions:

A mixed cold and warm standby system, where the switch succeeds with probability :

Find the hazard function:

Generate random numbers and compare with probability density:

Standby system where one component can fail while in standby, and a switch with a lifetime:

Compare the survival functions with different switch failure rates:

## Applications(2)

The lifetime of a component is exponentially distributed. To improve reliability, a second identical component is acquired. Find the most efficient use of this second component:

One alternative is a parallel configuration:

Another alternative is a standby configuration, with a switch that succeeds with probability p:

Plot the survival function of the two alternatives and compare with the original component, assuming perfect switching:

Simulate failure times for 30 standby systems and find the best configuration:

Check how bad a switch you can use while still being better than a parallel system:

The requirement on the switch to equal a parallel system gets lower with time:

Consider a computer server. It requires a power supply, hard drives, a network card, and a router to fulfill its intended function. The power supply is backed by a backup power outlet and a diesel generator in cold standby:

The hard drives are in a RAID configuration, which requires 2 out of 3 to work:

The network card has a second card in standby:

Two routers are connected in parallel:

The resulting survival function:

Plot it:

Compute the mean time to failure numerically:

Find the probability that the server survives for three months:

Define a consumer version that does not contain any redundancy:

Compare the survival functions:

## Properties & Relations(9)

Cold standby corresponds to the sum of component lifetimes:

Compare the survival functions:

Cold standby with identical exponentially distributed components is an ErlangDistribution:

Cold standby where component lifetimes follow the ExponentialDistribution corresponds to the HypoexponentialDistribution:

StandbyDistribution is a special case of TransformedDistribution:

Compare the survival functions:

StandbyDistribution is a special case of MixtureDistribution:

Compare the probability density function:

StandbyDistribution can be used in ReliabilityDistribution:

Compute the survival function:

ReliabilityDistribution can be used in StandbyDistribution:

Generate random numbers:

Compare with the probability density function:

StandbyDistribution can be used in FailureDistribution:

Compute the survival function:

FailureDistribution can be used in StandbyDistribution:

Generate random numbers:

Compare with the probability density function:

## Possible Issues(1)

Component distributions need to have a positive domain: Use TruncatedDistribution to restrict the domain to positive values only: