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 -> ({#[[1]], #[[2]], #[[3]], #[[4]]} &), 5 -> ({Max[{#[[1]], #[[2]]}], #[[3]], #[[4]], #[[5]]} &), 6 -> ({Max[{#[[1]], #[[2]]}], Max[{#[[3]], #[[4]]}], #[[5]], #[[6]]} &), 7 -> ({Max[{#[[1]], #[[2]]}], Max[{#[[3]], #[[4]]}], Max[#[[5]], #[[6]]], #[[7]]} &), 8 -> ({Max[{#[[1]], #[[2]]}], Max[{#[[3]], #[[4]]}], Max[#[[5]], #[[6]]], Max[#[[7]], #[[8]]]} &), 9 -> ({Max[{#[[1]], #[[2]], #[[3]]}], Max[{#[[4]], #[[5]]}], Max[#[[6]], #[[7]]], Max[#[[8]], #[[9]]]} &), 10 -> ({Max[{#[[1]], #[[2]], #[[3]]}], Max[{#[[4]], #[[5]], #[[6]]}], Max[#[[7]], #[[8]]], Max[#[[9]], #[[10]]]} &), 11 -> ({Max[{#[[1]], #[[2]], #[[3]]}], Max[{#[[4]], #[[5]], #[[6]]}], Max[#[[7]], #[[8]], #[[9]]], Max[#[[10]], #[[11]]]} &), 12 -> ({Max[{#[[1]], #[[2]], #[[3]]}], Max[{#[[4]], #[[5]], #[[6]]}], Max[#[[7]], #[[8]], #[[9]]], Max[#[[10]], #[[11]], #[[12]]]} &) |>, lifeTimes, simulation}, lifeTimes = RandomVariate[ExponentialDistribution[5], {1, var}]; lifeTimes = figs[var][#] & /@ lifeTimes; simulation = Min /@ lifeTimes; 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 🙂