$2.$ Reliability $($a$)[2$ marks$]$ Highly available systems like cloud systems target metrics like "five nines". What does "five nines" represent in terms of seconds of downtime per year? $($b$)[4$ Marks$]$ Assuming I have some widget that has a probability of independent failure rate of $2\%.$ How many replicas do I need to achieve "five nines"? Mandatory: Show your working! $($c$)[2$ marks$]$ How many $$nines$"$ do I get if I add one more replica? $($d$)[5$ Marks$]$ The procurement office at your company now find a cheaper widget supplier. That supplier is somewhat old fashioned and only rates their product in MTBF $-$ which they specify as $80,000$ hours. That sounds like plenty to the staff in the procurement office who order the new widgets. How many replicas of these new widgets do I need to achieve my five nines target? Do not use the ARR adjustment, nor the approximate version of the formulas $($don$\text{'}$t round OK$).$ Mandatory: Show your working! $($e$)$ Are you pleased or displeased with the decision procurement office from question $2.($d$)$ if: O $[2$ marks$]$ The widgets are $30\%$ cheaper? $($ii$[2$ marks$]$ The widgets are $60\%$ cheaper?