Pivoting the bridge

Topic: Reliability · more questions on this topic
‹ Structure of a k-out-of-n system Paths and cuts of the bridge ›

In the bridge with side paths 1–4 and 2–5 and cross link 3, the structure function has the form \varphi(\mathbf{x})=x_3 z_1+(1-x_3)z_2. Find z_1 and z_2.

Worked answer (click to show)

When the cross link 3 works, it joins the two middle nodes, and the bridge becomes a parallel pair \{1,2\} in series with a parallel pair \{4,5\}:

z_1=\bigl(1-(1-x_1)(1-x_2)\bigr)\bigl(1-(1-x_4)(1-x_5)\bigr).

When it has failed, only the two side paths remain, in parallel:

z_2=1-(1-x_1x_4)(1-x_2x_5).

So \varphi(\mathbf x)=x_3z_1+(1-x_3)z_2.