Assuming the water supply for a tall building relies on a pumping system, we aim to assess the reliability$-$based metrics of this system over the course of one continuous year $(365$ days$).$ It's important to note that a single pump is used throughout the day, with a spare pump of similar specifications becoming available on Day $200$ to the end of the operation for immediate replacement if the main pump fails. Throughout that year, the main pump experienced $7$ failures at various hours $(1500,2400,3200,4000,4700,6000,7800).$ The maintenance times required after each failure for the main pump are $50,45,45,55,60,60,$ and $55$ hours, respectively. Let$$s assume that the time required to replace the spare pump is negligible, and the spare pump itself fails after every $1000$ hours of operation, with equal $50-$hour repair cycles after each failure.

$($A$)$ Create a figure to represent Time to Failure $($TTF$),$ Time to Repair $($TTR$),$ Time Between Failures $($TBF$),$ failure times, and restart times. Also, indicate on that figure when the spare pump should be activated over the one$-$year operation? $(3$ pts$)$

$($B$)$ Calculate failure rates for each of the main and spare pumps without considering their repairment times. In this case, consider the operation of each pump independently. $(3$ pts$)$

$($C$)$ Disregarding the influence of the spare pump, calculate the following metrics for the main pump based on REALISTIC POLICY: Mean Time to Failures $($MTTF$),$ Mean Time to Repair $($MTTR$),$ Mean Time Between Failures $($MTBF$),$ and the availability percentage. $(5$ pts$)$

$($D$)$ Calculate water availability based on REALISTIC POLICY, considering both the main and spare pumps in the system. $(4$ pts$)$