# I. M. Sobol – The Monte Carlo Method – Section 06 – Reliability Simulation Suppose we want to estimate the mean duration of failure-proof operation of the apparatus provided all characteristics of the failure-proof operation of each of the elements are known. If we assume that the duration of failure-proof operation of each element t(k) is fixed, then the calculation of time t of failure-proof operation of the whole unit presents no problems. For example, failure of anyone of the elements of the unit drawn in Fig-18 entails the failure of the whole unit, i.e.

t = Min[ t(1), t(2), t(3), t(4) ]

For the unit shown in Fig-19 where one of the elements has a stand-by element

t  = Min[ t(1), t(2), Max[ t(3), t(4) ], t(5) ]

since when one element, N3 for example, will break down, the unit will continue functioning on the remaining element N4

Simulation

There are a bunch of ways we can arrive at this simulation. But before that, look at the above images from the textbook.

• Figure:18 shows four resistors connected in series. Let us define this connection as follows
• R~R~R~R alternatively 1-1-1-1
• Figure:19 shows 5 resistors connected such that 3 and 4 are in parallel connected to the other ones that are in series
• R~R~R|R~R alternatively 1-1-2-1
• Given that you understand this notation, we will run simulations for the following configurations
• R~R~R~R alternatively 1-1-1-1
• R|R~R~R~R alternatively 2-1-1-1
• R|R~R|R~R~R alternatively 2-2-1-1
• R|R~R|R~R|R~R alternatively 2-2-2-1
• R|R~R|R~R|R~R|R alternatively 2-2-2-2
• R|R|R~R|R~R|R~R|R alternatively 3-2-2-2
• R|R|R~R|R|R~R|R~R|R alternatively 3-3-2-2
• R|R|R~R|R|R~R|R|R~R|R alternatively 3-3-3-2
• R|R|R~R|R|R~R|R|R~R|R|R alternatively 3-3-3-3
• Also note that the study here is nothing to do with the actual resistances of the devices but only the lifetimes
• All the resistors have same lifetime distribution which is exponential with a λ = 5
• All the plots below represent the same data but with different chart styles. The names of the chart styles in sequence are
• Density
• HistogramDensity
• PointDensity
• SmoothDensity
• Notice how the reliability increases as we increase the number of resistors that have a “backup parallel resistor attached to it”
• which is further enhanced by the number of parallel resistors at each juntion    Code

```Column@Module[{plotData},
plotData = Table[
With[{var = #},
Module[{
figs = <|
4 -> ({#[], #[], #[], #[]} &),
5 -> ({Max[{#[], #[]}], #[], #[], #[]} &),

6 -> ({Max[{#[], #[]}],
Max[{#[], #[]}], #[], #[]} &),

7 -> ({Max[{#[], #[]}], Max[{#[], #[]}],
Max[#[], #[]], #[]} &),

8 -> ({Max[{#[], #[]}], Max[{#[], #[]}],
Max[#[], #[]], Max[#[], #[]]} &),

9 -> ({Max[{#[], #[], #[]}],
Max[{#[], #[]}], Max[#[], #[]],
Max[#[], #[]]} &),

10 -> ({Max[{#[], #[], #[]}],
Max[{#[], #[], #[]}], Max[#[], #[]],
Max[#[], #[]]} &),

11 -> ({Max[{#[], #[], #[]}],
Max[{#[], #[], #[]}],
Max[#[], #[], #[]], Max[#[], #[]]} &),

12 -> ({Max[{#[], #[], #[]}],
Max[{#[], #[], #[]}],
Max[#[], #[], #[]],
Max[#[], #[], #[]]} &)
|>,

RandomVariate[ExponentialDistribution, {1, var}];
Mean@simulation]] & /@ Range[4, 12], 2000];

DistributionChart[Transpose@plotData, ChartElementFunction -> #,
ChartLabels ->
Placed[Rotate[#,
0 \[Pi]] & /@ (StringJoin[# <> "\n"] & /@ {"1-1-1-1",
"  2-1-1-1", "2-2-1-1", "2-2-2-1", "2-2-2-2", "3-2-2-2",
"3-3-2-2", "3-3-3-2", "3-3-3-3"}), Above],
ImageSize -> 788,
PlotLabel ->
"Configurations and Lifetime Profiles"] & /@ {"SmoothDensity",
"PointDensity", "Density", "HistogramDensity"}]```

End of the post 🙂