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Simple structure function - interpreting answer [Probability & Reliability Theory]

  1. Oct 6, 2012 #1
    1. The problem statement, all variables and given/known data
    This isn't a homework question. I'm working through my book's exercises and am having difficulty interpreting an answer. Any guidance will be very much appreciated.

    The problem is to come up with a structure function for a graph (image attached with this post). The answer is [itex]\phi(x) = x_1 max (x_2, x_3, x_4) x_5[/itex]

    2. The attempt at a solution
    Two types of unbroken circuit are possible:

    i) There is a connection from [itex]x_1[/itex] to [itex]x_4[/itex] (e.g. [itex]x_1 → x_2 → x_4[/itex] OR [itex]x_1 → x_3 → x_2 → x_4[/itex])
    ii) The triangle [itex]x_1,x_2, x_3[/itex] is connected even if [itex]x_4[/itex] is broken. Hence, current can loop from [itex]x_1 → x_2 → x_3[/itex] or from [itex]x_1 → x_3 → x2.[/itex]

    In the diagram, I can see four nodes [itex]x_1,x_2, x_3, x_4[/itex]. Where does [itex]x_5[/itex] come from?

    The following part of the answer also confuses me:
    [itex]max (x_2, x_3, x_4)[/itex]

    Why is this the case? I can't see how the circuit would function if [itex]x_2[/itex] and [itex]x_3[/itex] were both broken. Current simply could not flow from [itex]x_1[/itex] to [itex]x_4[/itex] unless component 2 or 3 were also working.
     

    Attached Files:

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