Each one of three departments, Dept A$,$ Dept B and Dept C$,$ of a manufacturing factory has exactly one special TZ battery, and this battery needs re$-$charging from time to time. The three TZ batteries are identical, and the after$-$re$-$charge life of each battery has negative exponential distribution with a mean of eight $(8)$ hours. One battery re$-$charge shop serves the three departments, Dept A$,$ Dept B and Dept $C.$ Because the $TZ$ battery requires a special equipment for its re$-$charge, this battery re$-$charge shop can only re$-$charge one TZ battery at a time, and has battery storage space for up to two batteries that may be waiting to be re$-$charged. Assume that it takes an average of $60$ minutes to re$-$charge a TZ battery at this shop, and that battery re$-$charge time has negative exponential distribution. Assume also that re$-$charged batteries are promptly returned to the departments that brought them in$.$

$($a$)$ To keep track of number of batteries at the battery re$-$charge shop, formulate a CTMC in rate form.

$($b$)$ Let $G$ denote the set of states of your CTMC that correspond to the re$-$charge shop being idle. Compute $STC=s(G)+s(G^c),$ and indicate its usefulness.