A small engine repair shop is run by three mechanics. Early in March of each year,

people bring in their tillers and lawn mowers for service and maintenance. The shop

is willing to accept all the tillers and mowers that customers bring in$.$ However, when

new customers see the floor of the shop covered with waiting jobs, they go elsewhere

for more prompt service. The floor shop can house a maximum of $12$ mowers or tillers,

excluding those being serviced. The customers arrive at the shop every $15$ minutes on

the average, and it takes a mechanic an average of $40$ minutes to complete each job.

Both the interarrival and the service times are exponential. Determine the following:

$($a$)$ Average number of idle mechanics.

$($b$)$ Amount of business lost to competition per $8-$hour day because of the limited capacity

of the shop.