$2.$ Consider a computer system called $3$ P $2$ M $,$ which consists of three processors and two shared memories communicating over a shared bus, as shown in the following figure. The system is operational if at least two processors can communicate with at least one of the two memories over the bus.

a$.$ Construct the fault tree model of this system

b$.$ Find all the minimal cut sets $($show the top$-$down generation process$).$

c$.$ Assume all the components fail independently with the following failure probability: $\backslash ($ p$=0\mathrm{.}01\backslash )$ for each processor; $\backslash ($ m$=0\mathrm{.}05\backslash )$ for each memory; and $\backslash ($ b$=0\mathrm{.}02\backslash )$ for the bus.

Determine the unreliability of the entire $3$ P $2$ M system.

Hint: an efficient method is to evaluate the processor subsystem using the method discussed for computing the reliability of the TMR design.