A process plant has five identical generators, one online and the other four on standby. Data suggests generators $1,3,4,$ and $5$ provide the intended service without any switching failure. However, generator $2$ has a switching failure probability of $P.$ Please $($i$)$ draw the rate diagram for the stated system and $($ii$)$ write down the ordinary differential equations for each state. $($iii$)$ What is the probability that the first generator is available from initiation of operation to $500$ hours if the MTBF $=200$ hours and $MTTR=2$ hours, assuming that a repair crew is available to repair the first generator when it fails?

$4+6+5=15\%$