$($a$)$ Determine an expression for the system availability $\backslash (\backslash$mathrm$\{$A$\}\_\{\backslash$mathrm$\{$s$\}\}\backslash )$ as a function of $\backslash (\backslash$mathrm$\{$A$\}\_\{1\},\backslash$mathrm$\{$~A$\}\_\{2\},\backslash$mathrm$\{$~A$\}\_\{3\},\backslash$mathrm$\{$~A$\}\_\{4\}\backslash )$ for both Figure $($a$)$ and Figure $($b$)$ above.

$($b$)$ Given the data in the table below for each component determine the availability of each component.

$($c$)$ Determine the downtime per year of Figure $($a$)$ and $($b$)$ above using the availability values of the Table in part $($b$).$ Compare the results

$($d$)$ Given a budget to improve the availability of SINGLE component in Figure $($a$)$ which component would you select to have the greatest effect on the system availability? Repeat for Figure $($b$)$