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.

(c) Probability that the next arriving customer will be serviced by the shop.

(d) Probability that at least one of the mechanics will be idle.

(e) Average number of tillers or mowers awaiting service.

(f) A measure of the overall productivity of the shop.