Vacuum System Reliability in a Circular Arrangement

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.

Set system size and adjacency threshold:

In[1]:=

n=20;k=3;

Visualize the circular arrangement of 20 vacuum bulbs and mark three adjacent failures in red:

In[2]:=

failed={1,2,3};Graph[CycleGraph[n],VertexStyle(Thread[failedRed]),EdgeStyleGray,GraphStyle"CircularEmbedding",VertexLabels"Name",VertexSize.4,ImageSize400,FrameTrue,FrameTicksNone,PlotRangePaddingScaled[0.15]]

Out[2]=

Build the structure function for a ring:

In[3]:=

bexpr=And@@Map[Or@@#&,Partition[Array[

x

#

&,n],k,1,{-1,-1}]]

Out[3]=

(

x

19

||

x

20

||

x

1

)&&(

x

20

||

x

1

||

x

2

)&&(

x

1

||

x

2

||

x

3

)&&(

x

2

||

x

3

||

x

4

)&&(

x

3

||

x

4

||

x

5

)&&(

x

4

||

x

5

||

x

6

)&&(

x

5

||

x

6

||

x

7

)&&(

x

6

||

x

7

||

x

8

)&&(

x

7

||

x

8

||

x

9

)&&(

x

8

||

x

9

||

x

10

)&&(

x

9

||

x

10

||

x

11

)&&(

x

10

||

x

11

||

x

12

)&&(

x

11

||

x

12

||

x

13

)&&(

x

12

||

x

13

||

x

14

)&&(

x

13

||

x

14

||

x

15

)&&(

x

14

||

x

15

||

x

16

)&&(

x

15

||

x

16

||

x

17

)&&(

x

16

||

x

17

||

x

18

)&&(

x

17

||

x

18

||

x

19

)&&(

x

18

||

x

19

||

x

20

)

Assign component lifetime models:

In[4]:=

dists=Array[{

x

#

,ExponentialDistribution[1]}&,n];

Assemble the reliability model:

In[5]:=

ℛ=ReliabilityDistribution[bexpr,dists];

Plot the survival function:

In[6]:=

Plot[SurvivalFunction[ℛ,t]//Evaluate,{t,0,1}]

Out[6]=

Compute the mean time to failure:

In[7]:=

Mean[ℛ]//N

Out[7]=

0.48861

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