$($Problem $1\mathrm{.}12$ in Hwang's book$)$ Consider the System Availability $($A$)$ of a server cluster in terms of three parameters: namely the mean time to failure $($MTTF$),$ the mean time to repair $($MTTR$),$ and a regular maintenance time $($RMT$).$ The MTTF reflects the average uptime between two adjacent natural failures. The MTTR is the downtime due to natural failure. The RMT refers to scheduled down time for hardware$/$software maintenance or updates.

$($a$)$ Given a cloud system with a demanded availability $A=98\%.$ If the MTTF is known to be two years $($or $365242=17,520$ hours$)$ and the MTTR is known $24$ hours, what is the value of RMT in hours per month that you can schedule for this cloud system?

$($b$)$ Consider a cloud cluster of three servers. The cluster is considered available $($or acceptable with a satisfactory performance level$),$ if at least $k$ servers are operating normally where $k3.$ Assume that each server has an availability rate of $p($or a failure rate of $1-p).$ Derive a formula to calculate the total cluster availability $A($i$.$e$.,$ the probability that the cluster is available satisfactorily$).$ Note that $A$ is a function of $k$ and $p.$

$($c$)$ Given that each server has an availability $p=0\mathrm{.}98,$ what is the largest minimum number of servers that must be available to achieve a total cluster availability $A,$ which is higher than $96\%?$ You have to check the effect of all possible values of $k$ in part $($b$)$ in order to answer this question correctly.