$(1)$ A decision to purchase one of two machines is to be made. Machine $1$ has a processing rate of transactions$/$ hour and it costs B dollars$/$hour to operate; machine $2$ is twice as fast but costs twice as much to operate. Suppose that transactions arrive at the system according to a Poisson process of rate and that the transaction processing times are exponentially distributed. The total cost of the system is the operation cost plus a cost of A dollars for each hour a customer has to wait.

$($a$)$ Find expressions for the total cost per hour for each of the systems.

$($b$)$ If A$=$ B$/10,$ for what range of arrival rates is machine $1$ cheaper? Repeat for A $=10$B$.$