Surveillance Cameras

Cameras are arranged in an overlapping grid. All areas are covered until a 2-by-2 grid fails. Each camera has an expected lifetime of six years. Compare the reliability of this system to a system with only four cameras, with and without standby components.

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structure = 
  BooleanConsecutiveFunction[{2, 2}, {4, 4}][
   Array[Subscript[x, #1, #2] &, {4, 4}]];
\[ScriptCapitalD]camera = ExponentialDistribution[1/6];
\[ScriptCapitalF] = 
  FailureDistribution[structure, 
   Join @@ Array[{Subscript[
       x, #1, #2], \[ScriptCapitalD]camera} &, {4, 4}]];
\[ScriptCapitalF]nooverlap = 
  FailureDistribution[
   Subscript[x, 1] || Subscript[x, 2] || Subscript[x, 3] || Subscript[
    x, 4], {{Subscript[x, 1], \[ScriptCapitalD]camera}, {Subscript[x, 
     2], \[ScriptCapitalD]camera}, {Subscript[x, 
     3], \[ScriptCapitalD]camera}, {Subscript[x, 
     4], \[ScriptCapitalD]camera}}];
\[ScriptCapitalD]standby = 
  StandbyDistribution[\[ScriptCapitalD]camera, \
{\[ScriptCapitalD]camera, \[ScriptCapitalD]camera, \
\[ScriptCapitalD]camera}, 0.9];
\[ScriptCapitalF]standby = 
  FailureDistribution[
   Subscript[x, 1] || Subscript[x, 2] || Subscript[x, 3] || Subscript[
    x, 4], {{Subscript[x, 1], \[ScriptCapitalD]standby}, {Subscript[x,
      2], \[ScriptCapitalD]standby}, {Subscript[x, 
     3], \[ScriptCapitalD]standby}, {Subscript[x, 
     4], \[ScriptCapitalD]standby}}];

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