P$1.($a$)$ A multiprocessor system consists of a k processor. The system can provide service if at

least one processor remains operational. All other supporting components are highly reliable.

Assume that each processor has a failure rate of $($lambda$)$ and the time to failure is

exponentially distributed. Derive an expression reliability Rk$($t$)$ and MTTF for this k processor

system. For simplicity, you may denote the reliability of one processor as Rp$($t$)($and can drop t

for brevity if so desired$).$ Plot Rk$($t$)$ and Rp$($t$)$ for $=0\mathrm{.}002$ and k $=8.$

$($b$)$ In another application the system fails if any of the processors fails. Derive an expression for

reliability Rs$($t$)$ and MTTF of this k processor system for this application.