$9\mathrm{.}3$ Data Centers, Backhoes, and Bugs

Data centers alternate between "working" and "down." There are many reasons

why data centers can be down, but for the purpose of this problem we mention

only two reasons: $($i$)$ a backhoe accidentally dug up some cable, or $($ii$)$ a software

bug crashed the machines. Suppose that a data center that is working today will

be down tomorrow due to backhoe reasons with probability $\frac{1}{6},$ but will be down

tomorrow due to a software bug with probability $\frac{1}{4}.$ A data center that is down to$-$

day due to backhoe reasons will be up tomorrow with probability $1.$ A data center

that is down today due to a software bug will be up tomorrow with probability $\frac{3}{4}.$

$($a$)$ Draw a DTMC for this problem.

$($b$)$ Is your DTMC ergodic? Why or why not?

$($c$)$ Is your DTMC time$-$reversible? Why or why not?

$($d$)$ What fraction of time is the data center working?

$($e$)$ What is the expected number of days between backhoe failures?