Vehicles leaving North Portal, Saskatchewan arrive at US customs in Portal, North Dakota in accordance with a Poisson process. On average, $3$ vehicles per hour arrive at the border crossing. There to greet them is Bullwinkle the Moose on behalf of the United States of America. Inspector Bullwinkle processes vehicles with exponential service times averaging $15$ minutes. For some reason, people don't mind waiting in queue, or being inspected by a moose. However, once Bullwinkle starts an inspection, passengers get increasingly impatient. If an inspection takes longer than $15$ minutes, the driver of the vehicle involved sends a complaint to the United States Government! Assume that this process is in steady state.

$($a$)$ What is the expected number of complaints to the United States Government generated during a $24$ hour period? HINT: What is the probability that the driver of the next vehicle to arrive at the customs stop will complaint to the United States Government?

$($b$)$ Once the customs stop empty, Bullwinkle takes a coffee break. The coffee break ends the instant the next vehicle arrives. In a $24$ hour period, what is the expected number of coffee breaks taken by Inspector Bullwinkle?